Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence
aa r X i v : . [ m a t h . C T ] A p r TWO KINDS OF DERIVED CATEGORIES, KOSZUL DUALITY,AND COMODULE-CONTRAMODULE CORRESPONDENCE
LEONID POSITSELSKI
Contents
Introduction 11. Derived Category of DG-Modules 82. Derived Categories of DG-Comodules and DG-Contramodules 213. Coderived and Contraderived Categories of CDG-Modules 284. Coderived Category of CDG-Comodulesand Contraderived Category of CDG-Contramodules 475. Comodule-Contramodule Correspondence 596. Koszul Duality: Conilpotent and Nonconilpotent Cases 667. A ∞ -Algebras and Curved A ∞ -Coalgebras 858. Model Categories of DG-Modules,CDG-Comodules, and CDG-Contramodules 1069. Model Categories of DG-Algebras and CDG-Coalgebras 112Appendix A. Homogeneous Koszul Duality 123Appendix B. D –Ω Duality 131References 142 Introduction equence related to a bicomplex is that the familiar picture of two spectral sequencesconverging to the same limit splits in two halves when the bicomplex becomes infi-nite enough. The two spectral sequences essentially converge to the cohomology oftwo different total complexes. To obtains those, one takes infinite products in the“positive” direction along the diagonals and infinite direct sums in the “negative”direction (like in Laurent series). The two possible choices of the “positive” and“negative” directions give rise to the two completions. The word “essentially” hereis to be understood as “ignoring the delicate, but often manageable issues related tononexactness of the inverse limit”.0.2. This alternative between taking infinite direct sums and infinite products whenconstructing the total complex leads to the classical distinction between differentialderived functors of the first and the second kind [25, section I.4]. Roughly speaking,one can consider a DG-module either as a deformation of its cohomology or as adeformation of itself considered with zero differential; the spectral sequences related tothe former and the latter kind of deformations essentially converge to the cohomologyof the differential derived functors of the first and the second kind, respectively. Derived categories of the first and the second kind are intended to serve as thedomains of the differential derived functors of the first and the second kind. Thisdoes not always work as smoothly as one wishes; one discovers that, for technicalreasons, it is better to consider derived categories of the first kind for algebras andderived categories of the second kind for coalgebras . The distinction between thederived functors/categories of the first and the second kind is only relevant whencertain finiteness conditions no longer hold; this happens when one considers eitherunbounded complexes, or differential graded modules.Let us discuss the story of two derived categories in more detail. When the finite-ness conditions do hold, the derived category can be represented in two simple ways.It is both the quotient category of the homotopy category by the thick subcategory ofcomplexes with zero cohomology and the triangulated subcategory of the homotopycategory formed by the complexes of projective or injective objects. In the gen-eral case, this simple picture splits in two halves. The derived category of the firstkind is still defined as the quotient category of the homotopy category by the thicksubcategory of complexes (DG-modules, . . . ) with zero cohomology. It can be alsoobtained as a full subcategory of the homotopy category, but the description of thissubcategory is more complicated [50, 29, 9]. On the other hand, the derived categoryof the second kind is defined as the quotient category of the homotopy category bya thick subcategory with a rather complicated description. At the same time, it isequivalent to the full subcategory of the homotopy category formed by complexes(DG-comodules, DG-contramodules, . . . ) which become injective or projective whenconsidered without the differential. .3. The time has come to mention that there exist two kinds of module categoriesfor a coalgebra: besides the familiar comodules , there are also contramodules [15].Comodules can be thought of as discrete modules which are unions of their finite-dimensional subcomodules, while contramodules are modules where certain infinitesummation operations are defined. For example, the space of linear maps from acomodule to any vector space has a natural contramodule structure.The derived category of the first kind is what is known as just the derived category :the unbounded derived category, the derived category of DG-modules, etc. Thederived category of the second kind comes in two dual versions: the coderived andthe contraderived category . The coderived category works well for comodules, whilethe contraderived category is useful for contramodules. The classical notion of aDG-(co)algebra itself can be generalized in two ways; the derived category of thefirst kind is well-defined for an A ∞ -algebra, while the derived category of the secondkind makes perfect sense for a CDG-coalgebra .Other situations exist when derived categories of the second kind are well-behaved.One of them is that of a CDG-ring whose underlying graded ring has a finite homolog-ical dimension. In this case, the coderived and contraderived categories coincide. Inparticular, this includes the case of a CDG-algebra whose underlying graded algebra isfree. Such CDG-algebras can be thought of as strictly counital curved A ∞ -coalgebras ;CDG-modules over the former with free and cofree underlying graded modules cor-respond to strictly counital curved A ∞ -comodules and A ∞ -contramodules over thelatter. For a cofibrant associative DG-algebra, the derived, coderived, and contrade-rived categories of DG-modules coincide. Since any DG-algebra is quasi-isomorphicto a cofibrant one, it follows that the derived category of DG-modules over anyDG-algebra can be also presented as a coderived and contraderived category.The functors of forgetting the differentials, assigning graded (co/contra)modulesto CDG-(co/contra)modules, play a crucial role in the whole theory of derived cate-gories of the second kind. So it is helpful to have versions of these functors definedfor arbitrary DG-categories. An attempt to obtain such forgetful functors leads toa nice construction of an almost involution on the category of DG-categories. Therelated constructions for CDG-rings and CDG-coalgebras are important for the non-homogeneous quadratic duality theory, particularly in the relative case [48].0.4. Now let us turn to (derived) Koszul duality. This subject originates fromthe classical Bernstein–Gelfand–Gelfand duality (equivalence) between the boundedderived categories of finitely generated graded modules over the symmetric and ex-terior algebras with dual vector spaces of generators [8]. Attempting to generalizethis straightforwardly to arbitrary algebras, one discovers that many restricting con-ditions have to be imposed: it is important here that one works with algebras overa field, that the algebras and modules are graded, that the algebras are Koszul, that ne of them is finite-dimensional, while the other is Noetherian (or at least coherent)and has a finite homological dimension.The standard contemporary source is [7], where many of these restrictions areeliminated, but it is still assumed that everything happens over a semisimple basering, that the algebras and modules are graded, and that the algebras are Koszul.In [6], Koszulity is not assumed, but positive grading and semisimplicity of the basering still are. The main goal of this paper is to work out the Koszul duality forungraded algebras and coalgebras over a field, and more generally, differential gradedalgebras and coalgebras. In this setting, the Koszulity condition is less important,although it allows to obtain certain generalizations of the duality results. As to theduality over a base more general than a field, in this paper we only consider thespecial case of D –Ω duality, i. e., the duality between complexes of modules over thering of differential operators and (C)DG-modules over the de Rham (C)DG-algebraof differential forms (see 0.6). The ring of functions (or sections of the bundle ofendomorphisms of a vector bundle) is the base ring in this case. For a more generaltreatment of the relative situation, we refer the reader to [48, Chapter 11], where aversion of Koszul duality is obtained for a base coring over a base ring.The thematic example of nonhomogeneous Koszul duality over a field is the re-lation between complexes of modules over a Lie algebra g and DG-comodules overits standard homological complex. Here one discovers that, when g is reductive, thestandard homological complex with coefficients in a nontrivial irreducible g -modulehas zero cohomology—even though it is not contractible, and becomes an injectivegraded comodule when one forgets the differential. So one has to consider a versionof derived category of DG-comodules where certain acyclic DG-comodules survive ifone wishes this category to be equivalent to the derived category of g -modules. Thatis how derived categories of the second kind appear in Koszul duality [17, 37, 30].0.5. Let us say a few words about the homogeneous case. In the generality ofDG-(co)algebras, the homogeneous situation is distinguished by the presence of anadditional positive grading preserved by the differentials. Such a grading is well-known to force convergence of the spectral sequences, so there is no difference betweenthe derived categories of the first and the second kind in the homogeneous case.It is very essential here that the grading be indeed positive (or negative) on theDG-(co/contra)modules as well as the DG-(co)algebras, as one can see already inthe example of the duality between the symmetric and the exterior algebras in onevariable, S = k [ x ] and Λ = k [ ε ] /ε . The graded S -module M = k [ x, x − ] correspondsto the acyclic complex of Λ-modules K = ( · · · −→ Λ −→ Λ −→ · · · ) whose everyterm is Λ and every differential is the multiplication with ε .The acyclic, but not contractible complex K of projective and injective Λ-modulesprovides the simplest way to distinguish between the derived categories of the first nd the second kind. In derived categories of the first kind, it represents the zeroobject and is not adjusted to various derived functors, while in derived categories ofthe second kind, it is adjusted to derived functors and represents a nonzero object. Sothe S -module M has to be excluded from the category of modules under considerationfor a duality between conventional derived categories of S -modules and Λ-modules tohold. The positivity condition on the internal grading accomplishes that much in thehomogeneous case. All the stronger conditions on the gradings considered in [7, 6]are unnecessary for the purposes of establishing derived Koszul duality.0.6. Several attempts have been made in the literature [28, 5] to obtain an equiv-alence between the derived category of modules over the ring/sheaf of differentialoperators and an appropriately defined version of derived category of DG-modulesover the de Rham complex of differential forms. More generally, let X be a smoothalgebraic variety and E be a vector bundle on X with a global connection ∇ . LetΩ( X, E nd ( E )) be the sheaf of graded algebras of differential forms with coefficientsin the vector bundle E nd ( E ) of endomorphisms of E , d ∇ be the de Rham differentialin Ω( X, E nd ( E )) depending on the connection ∇ , and h ∇ ∈ Ω ( X, E nd ( E )) be thecurvature of ∇ . Then the triple consisting of the sheaf of graded rings Ω( X, E nd ( E )),its derivation d ∇ , and the section h ∇ is a sheaf of CDG-rings over X . The derivedcategory of modules over the sheaf of rings D X, E of differential operators acting inthe sections of the vector bundle E on X turns out to be equivalent to the code-rived category (and the contraderived category, when X is affine) of quasi-coherentCDG-modules over the sheaf of CDG-rings Ω( X, E nd ( E )). The assumption about theexistence of a global connection in E can be dropped (see subsection B.1 for details).0.7. Yet another very good reason for considering derived categories of the secondkind is that in their terms a certain relation between comodules and contramodulescan be established. Namely, the coderived category of CDG-comodules and the con-traderived category of CDG-contramodules over a given CDG-coalgebra are naturallyequivalent. We call this phenomenon the comodule-contramodule correspondence ; itappears to be almost as important as the Koszul duality.One can generalize the comodule-contramodule correspondence to the case ofstrictly counital curved A ∞ -comodules and A ∞ -contramodules over a curved A ∞ -coal-gebra by considering the derived category of the second kind for CDG-modules overa CDG-algebra whose underlying graded algebra is a free associative algebra.0.8. This paper can be thought of as an extended introduction to the mono-graph [48], as indeed, its key ideas precede those of [48] both historically and logically.It would had been all but impossible to invent the use of exotic derived categoriesfor the purposes of [48] if these were not previously discovered in the work presentedbelow. Nevertheless, most results of this paper are not covered by [48], since it is ritten in the generality of DG- and CDG-modules, comodules, and contramodules,while [48] deals with nondifferential semi(contra)modules most of the time. This pa-per is focused on Koszul duality, while the goal of [48] is the semi-infinite cohomology.The fact that exotic derived categories arise in Koszul duality was essentiallydiscovered by Hinich [22], whose ideas were developed by Lef`evre-Hasegawa [37,Chapitres 1 and 2]; see also Fløystad [17], Huebschmann [24], and Nicol´as [43]. Theterminology of “coderived categories” was introduced in Keller’s exposition [30]. How-ever, the definition of coderived categories in [37, 30] was not entirely satisfactory, inour view, in that the right hand side of the purported duality is to a certain extentdefined in terms of the left hand side (the approach to D -Ω duality developed in [5,section 7.2] had the same problem). This defect is corrected in the present paper.The analogous problem is present in the definitions of model category structureson the categories of DG-coalgebras by Hinich [22] and Lef`evre-Hasegawa [37]. Thisproved harder to do away with: we obtain various explicit descriptions of the distin-guished classes of morphisms of CDG-coalgebras independent of the Koszul duality,but the duality functors are still used in the proofs.In addition, we emphasize contramodules and CDG-coalgebras, whose role in thederived categories of the second kind and derived Koszul duality business does notseem to have been appreciated enough.0.9. Now let us describe the content of this paper in more detail. In Section 1 weobtain two semiorthogonal decompositions of the homotopy category of DG-modulesover a DG-ring, providing injective and projective resolutions for the derived categoryof DG-modules. We also consider flat resolutions and use them to define the derivedfunctor Tor for a DG-ring. Besides, we construct a t-structure on the derived cate-gory of DG-modules over an arbitrary DG-ring. In anticipation of the forthcomingpaper [49], we also discuss silly filtrations on the same derived category. This sectioncontains no new results; it is included for the sake of completeness of the exposition.The derived categories of DG-comodules and DG-contramodules and the differ-ential derived functors Cotor C,I and Coext IC of the first kind for a DG-coalgebra C are briefly discussed in Section 2, the proofs of the main results of this section beingpostponed to Sections 5 and 7. Partial results about injective and projective resolu-tions for the coderived and contraderived categories of a CDG-ring are obtained inSection 3. The finite homological dimension, Noetherian, coherent, and Gorensteincases are considered; in the former situation, a natural definition of the differentialderived functor Tor B,II of the second kind for a CDG-ring B is given. In addition,we construct an “almost involution” on the category of DG-categories.In Section 4 we construct semiorthogonal decompositions of the homotopy cate-gories of CDG-comodules and CDG-contramodules over a CDG-coalgebra, providinginjective and projective resolutions for the coderived category of CDG-comodules nd the contraderived category of CDG-contramodules. We also define the differ-ential derived functors Cotor, Coext, and Ctrtor for a CDG-coalgebra, and give asufficient condition for a morphism of CDG-coalgebras to induce equivalences of thecoderived and contraderived categories. The comodule-contramodule correspondencefor a CDG-coalgebra is obtained in Section 5.Koszul duality (or “triality”, as there are actually two module categories onthe coalgebra side) is studied in Section 6. Two versions of the duality theoremfor (C)DG-modules, CDG-comodules, and CDG-contramodules are obtained, onevalid for conilpotent CDG-coalgebras only and one applicable in the general case.We also construct an equivalence between natural localizations of the categories ofDG-algebras (with nonzero units) and conilpotent CDG-coalgebras.We discuss the derived categories of A ∞ -modules and the co/contraderived cat-egories of curved A ∞ -co/contramodules in Section 7. We explain the relation be-tween strictly unital A ∞ -algebras and coaugmented CDG-coalgebra structures ongraded tensor coalgebras, and use it to prove the standard results about strictlyunital A ∞ -modules. The similar approach to strictly counital curved A ∞ -coalgebrasyields the comodule-contramodule correspondence in the A ∞ case.Model category structures (of the first kind) for DG-modules over a DG-ringand model category structures (of the second kind) for CDG-comodules andCDG-contramodules over a CDG-coalgebra are constructed in Section 8. We alsoobtain model category structures of the first kind for DG-comodules and DG-contra-modules over a DG-coalgebra, and model category structures of the second kindfor CDG-modules over a CDG-ring in the finite homological dimension, Noether-ian, coherent, and Gorenstein cases. Quillen equivalences related to the comodule-contramodule correspondence and Koszul duality are discussed.We consider the model categories of DG-algebras and conilpotent CDG-coalgebrasin Section 9. More precisely, it turns out that the latter category has to be “fi-nalized” in order to make it a model category. We also discuss DG-modules overcofibrant DG-algebras. Conilpotent curved A ∞ -coalgebras and co/contranilpotentcurved A ∞ -co/contramodules over them are introduced.Homogeneous Koszul duality is worked out in Appendix A. The (more general)covariant and the (more symmetric) contravariant versions of the duality are consid-ered separately. The equivalence between the derived category of modules over thering/sheaf of differential operations acting in a vector bundle and the coderived cat-egory of CDG-modules over the corresponding de Rham CDG-algebra is constructedin Appendix B. A desription of the bounded derived category of coherent D -modulesin terms of coherent CDG-modules is also obtained. Acknowledgement.
The author is grateful to Michael Finkelberg for posing theproblem of constructing derived nonhomogeneous Koszul duality. I want to express y gratitude to Vladimir Voevodsky for very stimulating discussions and encour-agement, without which this work would probably never have been done. I alsobenefited from discussions with Joseph Bernstein, Victor Ginzburg, Amnon Neeman,Alexander Beilinson, Henning Krause, Maxim Kontsevich, Tony Pantev, AlexanderPolishchuk, Alexander Kuznetsov, Lars W. Christensen, Jan ˇSˇtov´ıˇcek, Pedro Nicol´as,and Alexander Efimov. I am grateful to Ivan Mirkovic, who always urged me to writedown the material presented below. I want to thank Dmitry Arinkin, who communi-cated the proof of Theorem 3.11.2 to me and gave me the permission to include it inthis paper. Most of the content of this paper was worked out when I was a Member ofthe Institute for Advanced Study, which I wish to thank for its hospitality. I am alsoindebted to the participants of an informal seminar at IAS, where I first presentedthese results in the Spring of 1999. The author was partially supported by P. Deligne2004 Balzan prize, an INTAS grant, and an RFBR grant while writing the paper up.Parts of this paper have been written when I was visiting the Institut des Hautes´Etudes Scientifiques, which I wish to thank for the excellent working conditions.1. Derived Category of DG-Modules
DG-rings and DG-modules. A DG-ring A = ( A, d ) is a pair consisting of anassociative graded ring A = L i ∈ Z A i and an odd derivation d : A −→ A of degree 1such that d = 0. In other words, it is supposed that d ( A i ) ⊂ A i +1 and d ( ab ) = d ( a ) b + ( − | a | ad ( b ) for a , b ∈ A , where | a | denotes the degree of a homogeneouselement, i. e., a ∈ A | a | .A left DG-module ( M, d M ) over a DG-ring A is a graded left A -module M = L i ∈ Z M i endowed with a differential d M : M −→ M of degree 1 compatible with thederivation of A and such that d M = 0. The compatibility means that the equation d M ( ax ) = d ( a ) x + ( − | a | ad M ( x ) holds for all a ∈ A and x ∈ M .A right DG-module ( N, d N ) over A is a graded right A -module N endowed with adifferential d N of degree 1 satisfying the equations d N ( xa ) = d N ( x ) a + ( − | x | xd ( a )and d N = 0, where x ∈ N | x | .Let L and M be left DG-modules over A . The complex of homomorphisms Hom A ( L, M ) from L to M over A is constructed as follows. The componentHom iA ( L, M ) consists of all homogeneous maps f : L −→ M of degree i such that f ( ax ) = ( − ) i | a | af ( x ) for all a ∈ A and x ∈ L . The differential in the complexHom A ( L, M ) is given by the formula d ( f )( x ) = d M ( f ( x )) − ( − | f | f ( d L ( x )). Clearly,one has d ( f ) = 0; for any composable morphisms of left DG-modules f and g onehas d ( f g ) = d ( f ) g + ( − | f | f d ( g ). or any two right DG-modules R and N over A , the complex of homomorphismsHom A ( R, N ) is defined by the same formulas as above and satisfies the same prop-erties, with the only difference that a homogeneous map f : R −→ N belonging toHom A ( R, N ) must satisfy the equation f ( xa ) = f ( x ) a for a ∈ A and x ∈ R .Let N be a right DG-module and M be a left DG-module over A . The tensorproduct complex N ⊗ A M is defined as the graded quotient group of the gradedabelian group N ⊗ Z M by the relations xa ⊗ y = x ⊗ ay for x ∈ N , a ∈ A , y ∈ M ,endowed with the differential given by the formula d ( x ⊗ y ) = d ( x ) ⊗ y +( − | x | x ⊗ d ( y ).For any two right DG-modules R and N and any two left DG-modules L and M thenatural map of complexes Hom A ( R, N ) ⊗ Z Hom A ( L, M ) −→ Hom Z ( R ⊗ A L, N ⊗ A M )is defined by the formula ( f ⊗ g )( x ⊗ y ) = ( − | g || x | f ( x ) ⊗ g ( y ). Here Z is consideredas a DG-ring concentrated in degree 0.For any DG-ring A , its cohomology H ( A ) = H d ( A ), defined as the quotient ofthe kernel of d by its image, has a natural structure of graded ring. For a leftDG-module M over A , its cohomology H ( M ) is a graded module over H ( A ); for aright DG-module N , its cohomology H ( N ) is a right graded module over H ( A ).A DG-algebra A over a commutative ring k is a DG-ring endowed with DG-ringhomomorphism k −→ A whose image is contained in the center of the algebra A , where k is considered as a DG-ring concentrated in degree 0; equivalently, aDG-algebra is a complex of k -modules with a k -linear DG-ring structure. Remark.
One can consider DG-algebras and DG-modules graded by an abeliangroup Γ different from Z , provided that Γ is endowed with a parity homomorphismΓ −→ Z / ∈ Γ, so that the differentials would have degree .In particular, one can take Γ = Z /
2, that is have gradings reduced to parities, orconsider fractional gradings by using some subgroup of Q consisting of rationals withodd denominators in the role of Γ. Even more generally, one can replace the parityfunction with a symmetric bilinear form σ : Γ × Γ −→ Z /
2, to be used in the super signrule in place of the product of parities; one just has to assume that σ ( , ) = 1 mod 2.All the most important results of this paper remain valid in such settings. The onlyexceptions are the results of subsections 3.4 and 4.3, where we consider boundedgrading.1.2. DG-categories. A DG-category is a category whose sets of morphisms arecomplexes and compositions are biadditive maps compatible with the gradings andthe differentials. In other words, a DG-category DG consists of a class of ob-jects, complexes of abelian groups Hom DG ( X, Y ), called the complexes of morphismsfrom X to Y , defined for any two objects X and Y , and morphisms of complexesHom DG ( Y, Z ) ⊗ Z Hom DG ( X, Y ) −→ Hom DG ( X, Z ), called the composition maps, de-fined for any three objects X , Y , and Z . The compositions must be associative and nit elements id X ∈ Hom DG ( X, X ) must exist; the equations d (id X ) = 0 then holdautomatically.For example, left DG-modules over a DG-ring A form a DG-category, which wewill denote by DG ( A – mod ). The DG-category of right DG-modules over A will bedenoted by DG ( mod – A ).A covariant DG-functor DG ′ −→ DG ′′ consists of a map between the classes ofobjects and (closed) morphisms between the complexes of morphisms compatible withthe compositions. A contravariant DG-functor is defined in the same way, except thatone has to take into account the natural isomorphism of complexes V ⊗ W ≃ W ⊗ V for complexes of abelian groups V and W that is given by the formula v ⊗ w ( − | v || w | w ⊗ v . (Covariant or contravariant) DG-functors between DG ′ and DG ′′ forma DG-category themselves. The complex of morphisms between DG-functors F and G is a subcomplex of the product of the complexes of morphisms from F ( X ) to G ( X )in DG ′′ taken over all objects X ∈ DG ′ ; the desired subcomplex is formed by all thesystems of morphisms compatible with all morphisms X −→ Y in DG ′ .For example, a DG-ring A can be considered as a DG-category with a single object;covariant DG-functors from this DG-category to the DG-category of complexes ofabelian groups are left DG-modules over A , while contravariant DG-functors betweenthe same DG-categories can be identified with right DG-modules over A .A closed morphism f : X −→ Y in a DG-category DG is an element ofHom DG ( X, Y ) such that d ( f ) = 0. The category whose objects are the objects of DG and whose morphisms are closed morphisms in DG is denoted by Z ( DG ).An object Y is called the product of a family of objects X α (notation: Y = Q α X α ) if a closed isomorphism of contravariant DG-functors Hom DG ( − , Y ) ≃ Q α Hom DG ( − , X α ) is fixed. An object Y is called the direct sum of a family ofobjects X α (notation: Y = L α X α ) if a closed isomorphism of covariant DG-functorsHom DG ( Y, − ) ≃ Q α Hom DG ( X α , − ) is fixed.An object Y is called the shift of an object X by an integer i (notation: Y = X [ i ]) ifa closed isomorphism of contravariant DG-functors Hom DG ( − , Y ) ≃ Hom DG ( − , X )[ i ]is fixed, or equivalently, a closed isomorphism of covariant DG-functors Hom DG ( Y, − ) ≃ Hom DG ( X, − )[ − i ] is fixed.An object Z is called the cone of a closed morphism f : X −→ Y (notation: Z = cone( f )) if a closed isomorphism of contravariant DG-functors Hom DG ( − , Z ) ≃ cone( f ∗ ), where f ∗ : Hom DG ( − , X ) −→ Hom DG ( − , Y ), is fixed, or equivalently, aclosed isomorphism of covariant DG-functors Hom DG ( Z, − ) ≃ cone( f ∗ )[ − f ∗ : Hom DG ( Y, − ) −→ Hom DG ( X, − ), is fixed.Let V be a complex of abelian groups and p : V −→ V be an endomorphism ofdegree 1 satisfying the Maurer–Cartan equation d ( p ) + p = 0. Then one can define anew differential on V by setting d ′ = d + p ; let us denote the complex so obtained by V ( p ). Let q ∈ Hom DG ( X, X ) be an endomorphism of degree 1 satisfying the equation ( q ) + q = 0. An object Y is called the twist of the object X with respect to q if aclosed isomorphism of contravariant DG-functors Hom DG ( − , X ) ≃ Hom DG ( − , Y )( q ∗ )is fixed, where q ∗ ( g ) = q ◦ g for any morphism g whose target is X , or equivalently, aclosed isomorphism of covariant DG-functors Hom DG ( Y, − ) ≃ Hom DG ( X, − )( − q ∗ ) isfixed, where q ∗ ( g ) = ( − | g | g ◦ q for any morphism g whose source is Y .As any representing objects of DG-functors, all direct sums, products, shifts, cones,and twists are defined uniquely up to a unique closed isomorphism. The direct sumof a finite set of objects is naturally also their product, and vice versa. Finite directsums, products, shifts, cones, and twists are preserved by any DG-functors. One canexpress the cone of a closed morphism f : X −→ Y as the twist of the direct sum Y ⊕ X [1] with respect to the endomorphism q induced by f .Here is another way to think about cones of closed morphisms in DG-categories.Let DG denote the category whose objects are the objects of DG and morphisms arethe (not necessarily closed) morphisms in DG of degree 0. Let X ′ −→ X −→ X ′′ bea triple of objects in DG with closed morphisms between them that is split exact in DG . Then X is the cone of a closed morphism X ′′ [ − −→ X ′ . Conversely, for anyclosed morphism X −→ Y in DG with the cone Z there is a natural triple of objectsand closed morphisms Y −→ Z −→ X [1], which is split exact in DG .Let DG be a DG-category with shifts, twists, and infinite direct sums. Let · · · −→ X n −→ X n − −→ · · · be a complex of objects of DG with closed differentials ∂ n .Then the differentials ∂ n induce an endomorphism q of degree 1 on the direct sum L n X n [ n ] satisfying the equations d ( q ) = 0 = q . The twist of this direct sum withrespect to this endomorphism is called the total object of the complex X • formed bytaking infinite direct sums and denoted by Tot ⊕ ( X • ). For a DG-category DG withshifts, twists, and infinite products, one can consider the analogous construction withthe infinite direct sum replaced by the infinite product Q n X n [ n ]. Thus one obtainsthe definition of the total object formed by taking infinite products Tot ⊓ ( X • ).For a finite complex X • , the two total objects coincide and are denoted simply byTot( X • ); this total object only requires existence of finite direct sums/products forits construction. Alternatively, the total objects Tot, Tot ⊕ , and Tot ⊓ can be definedas certain representing objects of DG-functors. The finite total object Tot can bealso expressed in terms of iterated cones, so it is well-defined whenever cones exist ina DG-category DG , and it is preserved by any DG-functors.A DG-functor DG ′ −→ DG ′′ is said to be fully faithful if it induces isomorphismsof the complexes of morphisms. A DG-functor is said to be an equivalence ofDG-categories if it is fully faithful and every object of DG ′′ admits a closed iso-morphism with an object coming from DG ′ . This is equivalent to existence of aDG-functor in the opposite direction for which both the compositions admit closedisomorphisms to the identity DG-functors. DG-functors F : DG ′ −→ DG ′′ and G : DG ′′ −→ DG ′ are said to be adjoint if for every objects X ∈ DG ′ and Y ∈ DG ′′ closed isomorphism of complexes Hom DG ′′ ( F ( X ) , Y ) ≃ Hom DG ′ ( X, G ( Y )) is givensuch that these isomorphisms commute with the (not necessarily closed) morphismsinduced by morphisms in DG ′ and DG ′′ .Let DG be a DG-category where (a zero object and) all shifts and cones exist. Thenthe homotopy category H ( DG ) is the additive category with the same class of objectsas DG and groups of morphisms given by Hom H ( DG ) ( X, Y ) = H (Hom DG ( X, Y )).The homotopy category is a triangulated category [12]. Shifts of objects and cones ofclosed morphisms in DG become shifts of objects and cones of morphisms in the trian-gulated category H ( DG ). Any direct sums and products of objects of a DG-categoryare also their directs sums and products in the homotopy category. Adjoint functorsbetween DG-categories induce adjoint functors between the corresponding categoriesof closed morphisms and homotopy categories.Two closed morphisms f , g : X −→ Y in a DG-category DG are called homotopic if their images coincide in H ( DG ). A closed morphism in DG is called a homotopyequivalence if it becomes an isomorphism in H ( DG ). An object of DG is called contractible if it vanishes in H ( DG ).All shifts, twists, infinite direct sums, and infinite direct products exist in theDG-categories of DG-modules. The homotopy category of (the DG-category of)left DG-modules over a DG-ring A is denoted by Hot ( A – mod ) = H DG ( A – mod );the homotopy category of right DG-modules over A is denoted by Hot ( mod – A ) = H DG ( mod – A ).1.3. Semiorthogonal decompositions.
Let H be a triangulated category and A ⊂ H be a full triangulated subcategory. Then the quotient category H / A is de-fined as the localization of H with respect to the multiplicative system of morphismswhose cones belong to A . The subcategory A is called thick if it coincides with thefull subcategory formed by all the objects of H whose images in H / A vanish. A trian-gulated subcategory A ⊂ H is thick if and only if it is closed under direct summandsin H [53, 39]. The following Lemma is essentially due to Verdier [52]; see also [3, 11]. Lemma.
Let H be a triangulated category and B , C ⊂ H be its full triangulatedsubcategories such that Hom H ( B, C ) = 0 for all B ∈ B and C ∈ C . Then thenatural maps Hom H ( B, X ) −→ Hom H / C ( B, X ) and Hom H ( X, C ) −→ Hom H / B ( X, C ) are isomorphisms for any objects B ∈ B , C ∈ C , and X ∈ H . In particular, thefunctors B −→ H / C and C −→ H / B are fully faithful. Furthermore, the followingconditions are equivalent: (a) B is a thick subcategory in H and the functor C −→ H / B is an equivalence oftriangulated categories; (b) C is a thick subcategory in H and the functor B −→ H / C is an equivalence oftriangulated categories; c) B and C generate H as a triangulated category, i. e., any object of H can beobtained from objects of B and C by iterating the operations of shift and cone; (d) for any object X ∈ H there exists a distinguished triangle B −→ X −→ C −→ B [1] with B ∈ B and C ∈ C (and in this case for any morphism X ′ −→ X ′′ in H there exists a unique morphism between any distinguishedtriangles of the above form for X ′ and X ′′ , so this triangle is unique up to aunique isomorphism and depends functorially on X ); (e) C is the full subcategory of H formed by all the objects C ∈ H such that Hom H ( B, C ) = 0 for all B ∈ B , and the embedding functor B −→ H has aright adjoint functor (which can be then identified with the localization functor H −→ H / C ≃ B ); (f) C is the full subcategory of H formed by all the objects C ∈ H such that Hom H ( B, C ) = 0 for all B ∈ B , B is a thick subcategory in H , and thelocalization functor H −→ H / B has a right adjoint functor; (g) B is the full subcategory of H formed by all the objects B ∈ H such that Hom H ( B, C ) = 0 for all C ∈ C , and the embedding functor C −→ H has aleft adjoint functor (which can be then identified with the localization functor H −→ H / B ≃ C ); (h) B is the full subcategory of H formed by all the objects B ∈ H such that Hom H ( B, C ) = 0 for all C ∈ C , C is a thick subcategory in H , and thelocalization functor H −→ H / C has a left adjoint functor. (cid:3) Projective resolutions.
A DG-module M is said to be acyclic if it is acyclicas a complex of abelian groups, i. e., H ( M ) = 0. The thick subcategory of thehomotopy category Hot ( A – mod ) formed by the acyclic DG-modules is denoted by Acycl ( A – mod ). The derived category of left DG-modules over A is defined as thequotient category D ( A – mod ) = Hot ( A – mod ) / Acycl ( A – mod ).A left DG-module L over a DG-ring A is called projective if for any acyclicleft DG-module M over A the complex Hom A ( L, M ) is acyclic. The full trian-gulated subcategory of
Hot ( A – mod ) formed by the projective DG-modules is de-noted by Hot ( A – mod ) proj . The following Theorem says, in particular, that the ho-motopy category H = Hot ( A – mod ) and its subcategories B = Hot ( A – mod ) proj and C = Acycl ( A – mod ) satisfy the equivalent conditions of Lemma 1.3, and so describesthe derived category D ( A – mod ). Theorem. (a)
The category
Hot ( A – mod ) proj is the minimal triangulated subcategoryof Hot ( A – mod ) containing the DG-module A and closed under infinite direct sums. (b) The composition of functors
Hot ( A – mod ) proj −→ Hot ( A – mod ) −→ D ( A – mod ) is an equivalence of triangulated categories.Proof. First notice that the category
Hot ( A – mod ) proj is closed under infinite directsums. It contains the DG-module A , since for any DG-module M over A there is a atural isomorphism of complexes of abelian groups Hom A ( A, M ) ≃ M . According toLemma 1.3, it remains to construct for any DG-module M a morphism f : F −→ M inthe homotopy category of DG-modules over A such that the DG-module F belongs tothe minimal triangulated subcategory containing the DG-module A and closed underinfinite direct sums, while the cone of the morphism f is an acyclic DG-module.When A is a DG-algebra over a field k , it suffices to consider the bar-resolution ofa DG-module M . It is a complex of DG-modules over A , and its total DG-moduleformed by taking infinite direct sums provides the desired DG-module F .Let us give a detailed construction in the general case. Let M be a DG-moduleover A . Choose a complex of free abelian groups M ′ together with a surjectivemorphism of complexes M ′ −→ M such that the cohomology H ( M ′ ) is also a freegraded abelian group and the induced morphism of cohomology H ( M ′ ) −→ H ( M ) isalso surjective. For example, one can take M ′ to be the graded abelian group with thecomponents freely generated by nonzero elements of the components of M , endowedwith the induced differential. Set F = A ⊗ Z M ′ ; then there is a natural closedsurjective morphism F −→ M of DG-modules over A and the induced morphism ofcohomology H ( F ) −→ H ( M ) is also surjective. Let K be the kernel of the morphism F −→ M (taken in the abelian category Z DG ( A – mod ) of DG-modules and closedmorphisms between them). Applying the same construction to the DG-module K inplace of M , we obtain the DG-module F , etc. Let F be the total DG-module of thecomplex of DG-modules · · · −→ F −→ F formed by taking infinite direct sums. Onecan easily check that the cone of the morphism F −→ M is acyclic, since the complex · · · −→ H ( F ) −→ H ( F ) −→ H ( M ) −→ · · · −→ F −→ F −→ M coming from the silly filtration of this complex of complexes).It remains to show that the DG-module F as an object of the homotopy categorycan be obtained from the DG-module A by iterating the operations of shift, cone, andinfinite direct sum. Every DG-module F n is a direct sum of shifts of the DG-module A and shifts of the cone of the identity endomorphism of the DG-module A . Denoteby X n the total DG-module of the finite complex of DG-modules F n −→ · · · −→ F .Then we have F = lim −→ X n in the abelian category Z DG ( A – mod ). So there is an exacttriple of DG-modules and closed morphisms 0 −→ L X n −→ L X n −→ F −→ X n −→ X n +1 split in DG ( A – mod ) , the above exact triple alsosplits in this additive category. Thus F is a cone of the morphism L X n −→ L X n in the triangulated category Hot ( A – mod ). (cid:3) Injective resolutions.
A left DG-module M over a DG-ring A is said to be injective if for any acyclic DG-module L over A the complex Hom A ( L, M ) is acyclic.The full triangulated subcategory of
Hot ( A – mod ) formed by the injective DG-modulesis denoted by Hot ( A – mod ) inj . or any right DG-module N over A and any complex of abelian groups V thecomplex Hom Z ( N, V ) has a natural structure of left DG-module over A with thegraded A -module structure given by the formula ( af )( n ) = ( − | a | ( | f | + | n | ) f ( na ).The following Theorem provides another semiorthogonal decomposition of thehomotopy category Hot ( A – mod ) and another description of the derived category D ( A – mod ). Theorem. (a)
The category
Hot ( A – mod ) inj is the minimal triangulated subcategoryof Hot ( A – mod ) containing the DG-module Hom Z ( A, Q / Z ) and closed under infiniteproducts. (b) The composition of functors
Hot ( A – mod ) inj −→ Hot ( A – mod ) −→ D ( A – mod ) is an equivalence of triangulated categories.Proof. The proof is analogous to that of Theorem 1.4. Clearly, the category
Hot ( A – mod ) inj is closed under infinite products. It contains the DG-moduleHom Z ( A, Q / Z ), since the complex Hom A ( L, Hom Z ( A, Q / Z )) ≃ Hom Z ( L, Q / Z ) isacyclic whenever the DG-module L is. To construct an injective resolution of aDG-module M , one can embed in into a complex of injective abelian groups M ′ sothat the cohomology H ( M ′ ) is also injective and H ( M ) also embeds into H ( M ′ ).For example, one can take the components of M ′ to be the products of Q / Z overall nonzero homomorphisms of abelian groups from the components of M to Q / Z .Take J = Hom Z ( A, M ′ ) and consider the induced injective morphism of DG-modules M −→ J . Set K = J /M , J − = Hom Z ( A, K ′ ), etc., and J = Tot ⊓ ( J • ). Then themorphism of DG-modules M −→ J has an acyclic cone and the DG-module J is iso-morphic in Hot ( A – mod ) to a DG-module obtained from Hom Z ( A, Q / Z ) by iteratingthe operations of shift, cone, and infinite product. (cid:3) Flat resolutions.
A right DG-module N over a DG-ring A is said to be flat if for any acyclic left DG-module M over A the complex N ⊗ A M is acyclic. Flatleft DG-modules over A are defined in the analogous way. The full triangulatedsubcategory of Hot ( A – mod ) formed by flat DG-modules is denoted by Hot ( A – mod ) fl .We denote the thick subcategory of acyclic right A -modules by Acycl ( mod – A ) ⊂ Hot ( mod – A ). The quotient category Hot ( mod – A ) / Acycl ( mod – A ) is called the de-rived category of right DG-modules over A and denoted by D ( mod – A ). The fulltriangulated subcategory of flat right DG-modules is denoted by Hot ( mod – A ) fl ⊂ Hot ( mod – A ).It follows from Theorems 1.4–1.5 and Lemma 1.3 that one can compute the rightderived functor Ext A ( L, M ) = Hom D ( A – mod ) ( L, M ) for left DG-modules L and M over a DG-ring A in terms of projective or injective resolutions. Namely, one hasExt A ( L, M ) ≃ H (Hom A ( L, M )) whenever L is a projective DG-module or M is aninjective DG-module over A . The following Theorem allows to define a left derived unctor Tor A ( N, M ) for a right DG-module N and a left DG-module M over A sothat it could be computed in terms of flat resolutions. Theorem. (a)
The functor
Hot ( A – mod ) fl / ( Acycl ( A – mod ) ∩ Hot ( A – mod ) fl ) −→ D ( A – mod ) induced by the embedding Hot ( A – mod ) fl −→ Hot ( A – mod ) is an equiva-lence of triangulated categories. (b) The functor
Hot ( mod – A ) fl / ( Acycl ( mod – A ) ∩ Hot ( mod – A ) fl ) −→ D ( mod – A ) in-duced by the embedding Hot ( mod – A ) fl −→ Hot ( mod – A ) is an equivalence of triangu-lated categories. The proof of Theorem is based on the following Lemma.
Lemma.
Let H be a triangulated category and A , F ⊂ H be full triangulated subcat-egories. Then the natural functor F / A ∩ F −→ H / A is an equivalence of triangulatedcategories whenever one of the following two conditions holds: (a) for any object X ∈ H there exists an object F ∈ F together with a morphism F −→ X in H such that a cone of that morphism belongs to A , or (b) for any object Y ∈ H there exists an object F ∈ F together with a morphism Y −→ F in H such that a cone of that morphism belongs to A .Proof of Lemma. It is clear that the functor F / A ∩ F −→ H / A is surjective on theisomorphism classes of objects under either of the assumptions (a) or (b). To provethat it is bijective on morphisms, represent morphisms in both quotient categoriesby fractions of the form X ←− X ′ −→ Y in the case (a) and by fractions of the form X −→ Y ′ ←− Y in the case (b). (cid:3) Proof of Theorem.
Part (a): first notice that any projective left DG-module M over aDG-ring A is flat. Indeed, one has Hom Z ( N ⊗ A M, Q / Z ) ≃ Hom A ( M, Hom Z ( N, Q / Z ))for any right DG-module N over A , so whenever N is acyclic, and consequentlyHom Z ( N, Q / Z ) is acyclic, the left hand side of this isomorphism is acyclic, too,and therefore N ⊗ A M is acyclic. So it remains to use Theorem 1.4 together withLemma 1.3 and the above Lemma. To prove part (b), switch the left and right sidesby passing to the DG-ring A op defined as follows. As a complex, A op is identified with A , while the multiplication in A op is given by the formula a op b op = ( − | a || b | ( ba ) op .Then right DG-modules over A are left DG-modules over A op and vice versa. (cid:3) Now let us define the derived functorTor A : D ( mod – A ) × D ( A – mod ) −−→ k – mod gr for a DG-algebra A over a commutative ring k , where k – mod gr denotes the cate-gory of graded k -modules. For this purpose, restrict the functor of tensor product ⊗ A : Hot ( mod – A ) × Hot ( A – mod ) −→ Hot ( k – mod ) to either of the full subcategories Hot ( mod – A ) fl × Hot ( A – mod ) or Hot ( mod – A ) × Hot ( A – mod ) fl and compose it with he cohomology functor H : Hot ( k – mod ) −→ k – mod gr . The functors so obtained fac-torize through the localizations D ( mod – A ) × D ( A – mod ) and the two induced derivedfunctors D ( mod – A ) × D ( A – mod ) −→ k – mod gr are naturally isomorphic to each other.Indeed, the tensor product N ⊗ A M by the definition is acyclic whenever one ofthe DG-modules N and M is acyclic, while the other one is flat. Let us check thatthe complex N ⊗ A M is acyclic whenever either of the DG-modules N and M issimultaneously acyclic and flat. Assume that N is acyclic and flat; choose a flatleft DG-module F over A together with a morphism of DG-modules F −→ A withan acyclic cone. Then the complex N ⊗ A F is acyclic, since N is acyclic; while themorphism N ⊗ A F −→ N ⊗ A M is a quasi-isomorphism, since N is flat.To construct an isomorphism of the two induced derived functors, it suffices tonotice that both of them are isomorpic to the derived functor obtained by restrictingthe functor ⊗ A to the full subcategory Hot ( mod – A ) fl × Hot ( A – mod ) fl . In other words,suppose that G −→ N and F −→ M are morphisms of DG-modules with acycliccones, where the right DG-module G and the left DG-module F are flat. Then thereare natural quasi-isomorphisms G ⊗ A M ←− G ⊗ A F −→ N ⊗ A F .1.7. Restriction and extention of scalars.
Let f : A −→ B be a morphism ofDG-algebras, i. e., a closed morphism of complexes preserving the multiplication.Then any DG-module over B can be also considered as a DG-module over A , whichdefines the restriction-of-scalars functor R f : Hot ( B – mod ) −→ Hot ( A – mod ). Thisfunctor has a left adjoint functor E f given by the formula E f ( M ) = B ⊗ A M anda right adjoint functor E f given by the formula E f ( M ) = Hom A ( B, M ) (wherethe DG-module structure on Hom A ( B, M ) is defined so that Hom A ( B, M ) −→ Hom Z ( B, M ) is a closed injective morphism of DG-modules).The functor R f obviously maps acyclic DG-modules to acyclic DG-modules, andso induces a functor D ( B – mod ) −→ D ( A – mod ), which we will denote by I R f . Thefunctor E f has a left derived functor L E f obtained by restricting E f to either ofthe full subcategories Hot ( A – mod ) proj or Hot ( A – mod ) fl ⊂ Hot ( A – mod ) and com-posing it with the localization functor Hot ( B – mod ) −→ D ( B – mod ). The functor E f has a right derived functor R E f obtained by restricting E f to the full subcate-gory Hot ( A – mod ) inj ⊂ Hot ( A – mod ) and composing it with the localization functor Hot ( B – mod ) −→ D ( B – mod ). The functor L E f is left adjoint to the functor I R f andthe functor R E f is right adjoint to the functor I R f . Theorem.
The functors I R f , L E f , R E f are equivalences of triangulated categoriesif and only if the morphism f induces an isomorphism H ( A ) ≃ H ( B ) .Proof. Morphisms in D ( A – mod ) between shifts of the DG-module A recover the co-homology H ( A ) and analogously for the DG-algebra B , so the “only if” assertionfollows from the isomorphism L E f ( A ) ≃ B . To prove the “if” part, we will showthat the adjunction morphisms L E f ( I R f ( N )) −→ N and M −→ I R f ( L E f ( M )) re isomorphisms for any left DG-modules M over A and N over B . The for-mer morphism is represented by the composition B ⊗ A G −→ B ⊗ A N −→ N for any flat DG-module G over A endowed with a quasi-isomorphism G −→ N ofDG-modules over A . This composition is a quasi-isomorphism, since the morphisms B ⊗ A G ←− A ⊗ A G −→ A ⊗ A N ≃ N are quasi-isomorphisms. The latter morphismis represented by the fraction M ←− F −→ B ⊗ A F for any flat DG-module F over A endowed with a quasi-isomorphism F −→ M of DG-modules over A . The morphism F ≃ A ⊗ A F −→ B ⊗ A F is a quasi-isomorphism. (cid:3) DG-module t-structure.
An object Y of a triangulated category D is calledan extension of objects Z and X if there is a distinguished triangle X −→ Y −→ Z −→ X [1]. Let D = D ( A – mod ) denote the derived category of left DG-modules overa DG-ring A . Let D > ⊂ D denote the full subcategory formed by all DG-modules M over A such that H i ( M ) = 0 for i < D ⊂ D denote the minimal fullsubcategory of D ( A – mod ) containing the DG-modules A [ i ] for i > Theorem. (a)
The pair of subcategories ( D , D > ) defines a t-structure [3] on thederived category D ( A – mod ) . (b) The subcategory D ⊂ D coincides with the full subcategory formed by allDG-modules M over A such such that H i ( M ) = 0 for i > if and only if H i ( A ) = 0 for all i > .Proof. Part (a): clearly, one has D [1] ⊂ D , D > [ − ⊂ D > , and Hom D ( D , D > [ − M over A a closed mor-phism of DG-modules F −→ M inducing a monomorphism on H and an isomor-phism on H i for all i F can be obtained from the DG-modules A [ i ]with i > M ′ −→ M onto M froma complex of free abelian groups M ′ with free abelian groups of cohomology so that H i ( M ′ ) = 0 for i > H i ( M ′ ) −→ H i ( M ) are surjective for all i F = A ⊗ Z M ′ and K = ker( F → M ) one chooses a surjective morphism K ′ −→ K onto K from a complex of free abelian groups K ′ with free abelian groupsof cohomology so that H i ( K ′ ) = 0 for i > H i ( K ′ ) −→ H i ( K ) aresurjective for all i F = A ⊗ Z K ′ , etc. The DG-module F is con-structed as the total DG-module of the complex · · · −→ F −→ F formed by takinginfinite direct sums. The “only if” assertion in part (b) is clear. To prove “if”, replace A with its quasi-isomorphic DG-subring τ A with the components ( τ A ) i = A i for i <
0, ( τ A ) = ker( A → A ), and ( τ A ) i = 0 for i >
0; then notice that thecanonical filtrations on DG-modules over τ A considered as complexes of abeliangroups are compatible with the action of the ring τ A . (cid:3) emark 1. The t-structure described in part (a) of Theorem can well be degenerate,though it is clearly nondegenerate under the assumptions of part (b). Namely, onecan have T i D [ i ] = 0. For example, take A = k [ x ] to be the graded algebra ofpolynomials with one generator x of degree 1 over a field k and endow it with thezero differential. Then the graded A-module k [ x, x − ] considered as a DG-modulewith zero differential belongs to the above intersection, since it can be presented asthe inductive limit of the DG-modules x − j k [ x ]. Moreover, take A = k [ x, x − ], wheredeg x = 1 and d ( x ) = 0; then D > = 0 and D = D . Remark 2.
One might wish to define a dual version of the above t-structure on D ( A – mod ) where D > would be the minimal full subcategory of D containing theDG-modules Hom Z ( A, Q / Z )[ i ] for i D would consist of all DG-modules M with H i ( M ) = 0 for i > Remark 3.
The above construction of the DG-module t-structure can be generalizedin the following way (cf. [51]). Let D be a triangulated category with infinite directsums. An object C ∈ D is said to be compact if the functor Hom D ( C, − ) preservesinfinite direct sums. Let C ⊂ D be a subset of objects of D consisting of compactobjects and such that C [1] ⊂ C . Let D > be the full subcategory of D formed byall objects X such that Hom D ( C, X [ − C ∈ C , and let D be theminimal full subcategory of D containing C and closed under extensions and infinitedirect sums. Then ( D , D > ) is a t-structure on D . Indeed, let X be an object of D .Consider the natural map into X from the direct sum of objects from C indexedby morphisms from objects of C to X ; let X be the cone of this map. Applyingthe same construction to the object X in place of X , we obtain the object X , etc.Let Y be the homotopy inductive limit of X n , i. e., the cone of the natural map L n X n −→ L n X n . Then Y ∈ D > [ −
1] and cone( X → Y )[ − ∈ D .1.9. Silly filtrations.
Let A be a DG-ring and D = D ( A – mod ) denote the derivedcategory of left DG-modules over it. Denote by D ⊂ D the full subcategory formedby all the DG-modules M such that H i ( M ) = 0 for i > D > ⊂ D the fullsubcategory of all the DG-modules M such that H i ( M ) = 0 for i Theorem.
One has H i ( A ) = 0 for all i < if and only if any object X ∈ D can beincluded into a distinguished triangle Y −→ X −→ Z −→ Y [1] in D with Y ∈ D > and Z ∈ D . roof. “Only if”: suppose that the left DG-module X = A over A can be includedinto a distinguished triangle Y −→ X −→ Z −→ Y [1] in D with Y ∈ D > and Z ∈ D − (in the obvious notation). Then Hom D ( X, Z ) ≃ H ( Z ) = 0, hence X is adirect summand of Y in D , so H i ( X ) = 0 for i < M be a DG-module representing the object X . Let P be the direct sumof left DG-modules A [ − i ] over A taken over all the elements x ∈ H i ( M ) for all i > P −→ M ; set M to be its cone.One has H i ( P ) = 0 for all i H i ( P ) −→ H i ( M ) are surjective forall i >
1. Hence the maps H i ( M ) −→ H i ( M ) are isomorphisms for all i <
0, amonomorphism for i = 0, and zero maps for i >
0. Repeating the same procedurefor the DG-module M , etc., we obtain a sequence of DG-module morphisms M −→ M −→ M −→ · · · , each morphism having the properties described above. Set N = lim −→ M i ; then there is a closed morphism of DG-modules M −→ N over A suchthat the maps H i ( M ) −→ H i ( N ) are isomorphisms for all i < i = 0, and one has H i ( N ) = 0 for all i >
0. Set Q = cone( M → N )[ − H i ( Q ) = 0 for all i
0. This provides the desired distinguished triangle. (cid:3)
Remark 1.
There is a much simpler proof of the above Theorem applicable in thecase when A i = 0 for all i <
0. However, unlike in the situation of (the proofof) Theorem 1.8(b), it is not true that any DG-ring with zero cohomology in thenegative degrees can be connected by a chain of quasi-isomorphisms with a DG-ringwith zero components in the negative degrees. For a counterexample, consider the freeassociative algebra A over a field k generated by the elements x , y , and η in degree 0, z in degree 1, and ξ in degree −
1, with the differential given by d ( x ) = d ( y ) = d ( z ) = 0, d ( ξ ) = xy , and d ( η ) = yz . One can check that H i ( A ) = 0 for i <
0. The nontrivialMassey product ξz − xη of the cohomology classes x , y , and z provides the obstruction. Erratum added five years later:
The counterexample in Remark 1 does not work.The Massey product of x , y , and z is indeed defined and does not contain zero in H ∗ ( A ), but it does not obstruct existence of a DG-algebra B quasi-isomorphic to A with B i = 0 for i <
0. In fact, the quotient algebra of A by the two-sided idealgenerated by ξ and d ( ξ ) can be used in the role of B . To the best of the author’sknowledge, the question whether any DG-ring with zero cohomology in the negativedegrees can be connected by a chain of quasi-isomorphisms with a DG-ring with zerocomponents in the negative degrees remains open. Remark 2.
The construction of the above Theorem can be extended to arbitraryDG-rings in the following way. Given a DG-ring A , denote by D ⊂ D = D ( A – mod )the full subcategory formed by all the DG-modules M over A such that H i ( M ) = 0 for i < D > ⊂ D the minimal full subcategory of D containing the DG-modules A [ − i ] with i > X i ∈ D > and a sequence f distinguished triangles Y i −→ Y i +1 −→ X i +1 −→ Y i [1] in D with Y = X , thehomotopy colimit cone( L i Y i → L i Y i ) of Y i should also belong to D > . Then thesame construction as in the above proof provides for any X ∈ D a distinguishedtriangle Y −→ X −→ Z −→ Y [1] with Y ∈ D > and Z ∈ D . This can begeneralized even further in the spirit of Remark 1.8.3, by considering an arbitrarytriangulated category D admitting infinite direct sums, and a set of compact objects C ⊂ D such that C [ − ⊂ C in the role of the DG-modules A [ − i ] with i >
1. Thesubcategory D > is then generated by C using the operations of infinite direct sum andcountably iterated extension, and the subcategory D consists of all object X ∈ D such that Hom D ( C, X ) = 0 for all C ∈ C . If one does not insist on the subcategories D and D > being closed under shifts in the respective directions, the condition that C [ − ⊂ C can be dropped. Moreover, the subcategories D and D > described inthis remark have a semiorthogonality property, Hom D ( Y, Z ) = 0 for all Y ∈ D > and Z ∈ D (cf. [45]). Indeed, given an object Z in a triangulated category D , the classof all objects Y ∈ D such that Hom D ( Y, Z ) = 0 is closed under infinite direct sumsand countably iterated extensions, as one can see using the fact that the first derivedfunctor of projective limit of a sequence of surjective maps of abelian groups vanishes.2.
Derived Categories of DG-Comodules and DG-Contramodules
Graded comodules.
Let k be a fixed ground field. A graded coalgebra C over k is a graded k -vector space C = L i ∈ Z C i endowed with a comultiplicationmap C −→ C ⊗ k C and a counit map C −→ k , which must be homogeneous linearmaps of degree 0 satisfying the coassociativity and counity equations. Namely, thecomultiplication map must have equal compositions with the two maps C ⊗ k C ⇒ C ⊗ k C ⊗ k C induced by the comultiplication map, while the compositions of thecomultiplication map with the two maps C ⊗ k C −→ C induced by the counit mapmust coincide with the identity endomorphism of C .A graded left comodule M over C is a graded k -vector space M = L i ∈ Z M i endowedwith a left coaction map M −→ C ⊗ k M , which must be a homogeneous linear map ofdegree 0 satisfying the coassociativity and counity equations. Namely, the coactionmap must have equal compositions with the two maps C ⊗ k M ⇒ C ⊗ k C ⊗ k M induced by the comultiplication map and the coaction map, while the compositionof the coaction map with the map C ⊗ k M −→ M induced by the counit map mustcoincide with the identity endomorphism of M . A graded right comodule N over C isa graded vector space endowed with a right coaction map N −→ N ⊗ k C satisfyingthe analogous linearity, homegeneity, coassociativity, and counity equations.The cotensor product of a graded right C -comodule N and a graded left C -comodule M is the graded vector space N (cid:3) C M defined as the kernel of the air of linear maps N ⊗ k M ⇒ N ⊗ k C ⊗ k M , one of which is induced by the rightcoaction map and the other by the left coaction map. There are natural isomorphisms C (cid:3) C M ≃ M and N (cid:3) C C ≃ N for any graded left C -comodule M and graded right C -comodule N .Graded left C -comodules of the form C ⊗ k V , where V is a graded vector space,are called cofree graded left C -comodules; analogously for graded right C -comodules.The category of graded C -comodules is an abelian category with enough injectives;injective graded C -comodules are exactly the direct summands of cofree graded C -comodules.For any graded left C -comodules L and M , the graded vector space Hom C ( L, M )consists of homogeneous linear maps f : L −→ M satisfying the condition that thecoaction maps of L and M form a commutative diagram together with the map f andthe map f ∗ : C ⊗ k L −→ C ⊗ k M given by the formula f ∗ ( c ⊗ x ) = ( − | f || c | c ⊗ f ( x ).For any graded right C -comodules R and N , the graded vector space Hom C ( R, N )consists of homogeneous linear maps f : R −→ N such that the coaction maps of R and N form a commutative diagram together with the map f and the map f ∗ : R ⊗ k C −→ N ⊗ k C given by the formula f ∗ ( x ⊗ c ) = f ∗ ( x ) ⊗ c . For any left C -comodule L and any graded vector space V there is a natural isomorphism Hom C ( L, C ⊗ k V ) ≃ Hom k ( L, V ); analogously in the right comodule case.2.2.
Graded contramodules. A graded left contramodule P over a graded coalge-bra C is a graded k -vector space P = L i ∈ Z P i endowed with the following structure.Let Hom k ( C, P ) be the graded vector space of homogeneous linear maps C −→ P ;then a homogeneous linear map Hom k ( C, P ) −→ P of degree 0, called the left con-traaction map, must be given and the following contraassociativity and counity equa-tions must be satisfied. For any graded vector spaces V , W , and P , define thenatural isomorphism Hom k ( V ⊗ k W, P ) ≃ Hom k ( W, Hom k ( V, P )) by the formula f ( w )( v ) = ( − | w || v | f ( v ⊗ w ). The comultiplication and the contraaction maps in-duce a pair of maps Hom k ( C ⊗ k C, P ) ≃ Hom k ( C, Hom k ( C, P )) ⇒ Hom k ( C, P ).These maps must have equal compositions with the contraaction map; besides, thecomposition of the map P −→ Hom k ( C, P ) induced by the counit map with thecontraaction map must coincide with the identity endomorphism of P .The graded vector space of cohomomorphisms Cohom C ( M, P ) from a graded left C -comodule M to a graded left C -contramodule P is defined as the cokernel of thepair of linear maps Hom k ( C ⊗ k M, P ) −→ Hom k ( M, P ), one of which is induced bythe left coaction map and the other by the left contraaction map. For any graded C -contramodule P there is a natural isomorphism Cohom C ( C, P ) ≃ P .For any graded right C -comodule N and any graded vector space V there is a natu-ral graded left C -contramodule structure on the graded vector space of homogeneouslinear maps Hom k ( N, V ) given by the left contraaction map Hom k ( C, Hom k ( N, V )) ≃ om k ( N ⊗ k C, V ) −→ Hom k ( N, V ) induced by the right coaction map. For anygraded left C -comodule M , graded right C -comodule N , and graded vector space V ,there is a natural isomorphism Hom k ( N (cid:3) C M, V ) ≃ Cohom C ( M, Hom k ( N, V )).Graded left C -contramodules of the form Hom k ( C, V ) are called free graded left C -contramodules. The category of graded left C -contramodules is an abelian categorywith enough projectives; projective graded left C -contramodules are exactly the directsummands of free graded left C -contramodules.The contratensor product of a graded right C -comodule N and a graded left C -contramodule P is the graded vector space N ⊙ C P defined as the cokernel ofthe pair of linear maps N ⊗ k Hom k ( C, P ) −→ N ⊗ k P , one of which is induced bythe left contraaction map, while the other one is obtained as the composition of themap induced by the right coaction map and the map induced by the evaluation map C ⊗ k Hom k ( C, P ) −→ P given by the formula c ⊗ f ( − | c || f | f ( c ). For any gradedright C -comodule N and any graded vector space V there is a natural isomorphism N ⊙ C Hom k ( C, V ) ≃ N ⊗ k V .For any graded left C -contramodules P and Q , the graded vector space Hom C ( P, Q )consists of all homogeneous linear maps f : P −→ Q satisfying the condition that thecontraaction maps of P and Q form a commutative diagram together with the map f and the map f ∗ : Hom k ( C, P ) −→ Hom k ( C, Q ) given by the formula f ∗ ( g ) = f ◦ g . Forany graded left C -contramodule Q and any graded vector space V there is a naturalisomorphism Hom C (Hom k ( C, V ) , Q ) ≃ Hom k ( V, Q ). For any right C -comodule N ,any graded left C -contramodule P , and any graded vector space V , there is a naturalisomorphism Hom k ( N ⊙ C P, V ) ≃ Hom C ( P, Hom k ( N, V )).The proofs of the results of this subsection are not difficult; some details can befound in [48]. The assertions stated in the last paragraph can be used to deduce theassertions of the preceding two paragraphs.
Remark.
Ungraded contramodules over ungraded coalgebras can be simply definedas graded contramodules concentrated in degree 0 over graded coalgebras concen-trated in degree 0. One might wish to have a forgetful functor assigning ungradedcontramodules over ungraded coalgebras to graded contramodules over graded coal-gebras. The construction of such a functor is delicate in two ways. Firstly, to assignan ungraded contramodule to a graded contramodule P , one has to take the directproduct of its grading components Q i ∈ Z P i rather than the direct sum, while theungraded coalgebra corresponding to a graded coalgebra C is still constructed as thedirect sum L i ∈ Z C i . Analogously, to assign an ungraded comodule to a graded co-module M one takes the direct sum L i ∈ Z M i , to assign an ungraded ring to a gradedring A one takes the direct sum L i ∈ Z A i , while to assign an ungraded module to agraded module M one can take either the direct sum L i ∈ Z M i , or the direct product Q i ∈ Z M i . Secondly, there is a problem of signs in the contraasociativity equation, hich is unique to graded contramodules (no signs are present in the definitions ofgraded algebras, modules, coalgebras, or comodules); it is resolved as follows. A mor-phism between graded vector spaces f : V −→ W , when it is not necessarily even,can be thought of either as a left or as a right morphism. The left and the rightmorphisms correspond to each other according to the sign rule f ( x ) = ( − | f || x | ( x ) f ,where f ( x ) is the notation for the left morphisms and ( x ) f for the right morphisms.The above definition of graded left contramodules P is given in terms of left mor-phisms C −→ P ; to define the functor of forgetting the grading, one has to reinterpretit in terms of right morphisms. The exposition in [48, Chapter 11] presumes right morphisms in this definition (even if the notation f ( x ) is being used from time totime).2.3. DG-comodules and contramodules. A DG-coalgebra C over a field k is agraded coalgebra endowed with a differential d : C −→ C of degree 1 with d = 0such that the comultiplication map C −→ C ⊗ k C and the counit map C −→ k aremorphisms of complexes. Here the differential on C ⊗ k C is defined as on the tensorproduct of two copies of the complex C , while the differential on k is trivial.A left DG-comodule M over a DG-coalgebra C is a graded comodule M over thegraded coalgebra C together with a differential d : M −→ M of degree 1 with d = 0such that the left coaction map M −→ C ⊗ k M is a morphism of complexes. Here C ⊗ k M is considered as the tensor product of the complexes C and M over k . RightDG-comodules are defined in the analogous way. A left DG-contramodule P over C isa graded contramodule P over C endowed with a differential d : P −→ P of degree 1with d = 0 such that the left contraaction map Hom k ( C, P ) −→ P is a morphismof complexes. Here Hom k ( C, P ) is endowed with the differential of the complex ofhomomorphisms from the complex C to the complex P over k .Whenever N is a right DG-comodule and M is a left DG-comodule over aDG-coalgebra C , the cotensor product N (cid:3) C M of the graded comodules N and M over the graded coalgebra C is endowed with the differential of the subcomplex ofthe tensor product complex N ⊗ k M . Whenever M is a left DG-comodule and P isa left DG-contramodule over a DG-coalgebra C , the graded vector space of cohomo-morphisms Cohom C ( M, P ) is endowed with the differential of the quotient complexof the complex of homomorphisms Hom k ( M, P ).Whenever N is a right DG-comodule and P is a left DG-contramodule over aDG-coalgebra C , the contratensor product N ⊙ C P of the graded comodule N and thegraded contramodule P over the graded coalgebra C is endowed with the differentialof the quotient complex of the tensor product complex N ⊗ k P .For any left DG-comodules L and M over a DG-coalgebra C , the graded vectorspace of homomorphisms Hom C ( L, M ) between the graded comodules L and M overthe graded coalgebra C is endowed with the differential of the subcomplex of thecomplex of homomorphisms Hom k ( L, M ). Differentials on the graded vector spaces f homomorphisms Hom C ( R, N ) and Hom C ( P, Q ) for right DG-comodules R , N andleft DG-contramodules P , Q over a DG-coalgebra C are constructed in the com-pletely analogous way. These constructions define the DG-categories DG ( C – comod ), DG ( comod – C ), and DG ( C – contra ) of left DG-comodules, right DG-comodules, andleft DG-contramodules over C , respectively.All shifts, twists, infinite direct sums, and infinite direct products exist in theDG-categories of DG-comodules and DG-contramodules. The homotopy category of(the DG-category of) left DG-comodules over C is denoted by Hot ( C – comod ), thehomotopy category of right DG-comodules over C is denoted by Hot ( comod – C ), andthe homotopy category of left DG-contramodules over C is denoted by Hot ( C – contra ).2.4. Injective and projective resolutions.
A DG-comodule M or a DG-contra-module P is said to be acyclic if it is acyclic as a complex of vector spaces, i. e., H ( M ) = 0 or H ( P ) = 0, respectively. The classes of acyclic DG-comodules over aDG-coalgebra C are closed under shifts, cones, and infinite direct sums, while theclass of acyclic DG-contramodules is closed under shifts, cones, and infinite products.The thick subcategories of the homotopy categories Hot ( C – comod ), Hot ( comod – C ),and Hot ( C – contra ) formed by the acyclic DG-comodules and DG-contramodulesover C are denoted by Acycl ( C – comod ), Acycl ( comod – C ), and Acycl ( C – contra ), re-spectively. The derived categories of left DG-comodules, right DG-comodules, andleft DG-contramodules over C are defined as the quotient categories D ( C – comod ) = Hot ( C – comod ) / Acycl ( C – comod ), D ( comod – C ) = Hot ( comod – C ) / Acycl ( comod – C ),and D ( C – contra ) = Hot ( C – contra ) / Acycl ( C – contra ).A left DG-comodule M over a DG-coalgebra C is called injective if for any acyclicleft DG-comodule L over C the complex Hom C ( L, M ) is acyclic. The full triangulatedsubcategory of
Hot ( C – comod ) formed by the injective DG-comodules is denoted by Hot ( C – comod ) inj . A left DG-contramodule P over a DG-coalgebra C is called pro-jective if for any acyclic left DG-contramodule Q over C the complex Hom C ( P, Q ) isacyclic. The full triangulated subcategory of
Hot ( C – contra ) formed by the projectiveDG-contramodules is denoted by Hot ( C – contra ) proj . Theorem. (a)
The composition of functors
Hot ( C – comod ) inj −→ Hot ( C – comod ) −→ D ( C – comod ) is an equivalence of triangulated categories. (b) The composition of functors
Hot ( C – contra ) proj −→ Hot ( C – contra ) −→ D ( C – contra ) is an equivalence of triangulated categories. Proof will be given in subsection 5.5.
Remark.
The analogue of Theorem 1.7 does not hold for DG-coalgebras. Moreprecisely, given a morphism of DG-coalgebras f : C −→ D , any DG-comodule over C can be considered as a DG-comodule over D and any DG-contramodule over C can beconsidered as a DG-contramodule over D , so there are restriction-of-scalars functors R f : D ( C – comod ) −→ D ( D – comod ) and I R f : D ( C – contra ) −→ D ( D – contra ). Usingthe above Theorem, one can construct the functor R E f right adjoint to I R f andthe functor L E f left adjoint to I R f (cf. 4.8). Just as in the proof of Theorem 1.7,one can show that the morphism f induces an isomorphism of cohomology whenevereither of the functors I R f or I R f is an equivalence of categories. However, there existquasi-isomorphisms of DG-coalgebras f : C −→ D for which the functors I R f and I R f are not equivalences. The following counterexample is due to D. Kaledin [27].Let E and F be DG-coalgebras and K be a DG-bicomodule over E and F , i. e.,a complex of k -vector spaces with commuting structures of left DG-comodule over E and right DG-comodule over F . Then there are natural DG-coalgebra structureson the direct sums C = E ⊕ K ⊕ F and D = E ⊕ F ; there are also morphisms ofDG-coalgebras D −→ C −→ D . When the complex K is acyclic, these morphisms arequasi-isomorphisms. A left DG-comodule over C is the same that a pair ( M, N ) of leftDG-comodules M over E and N over F together with a closed morphism M −→ K (cid:3) F N of left DG-comodules over E . For a DG-comodule M over E and a DG-comodule N over F , morphisms ( M, −→ (0 , N [1]) in the derived category D ( C – comod ) arerepresented by morphisms M ′ −→ K (cid:3) F N ′ in the homotopy category Hot ( E – comod )together with quasi-isomorphisms of DG-comodules M ′ −→ M over E and N −→ N ′ over F . In particular, when M is a projective DG-comodule (i. e., there are nononzero morphisms in Hot ( E – comod ) from M into acyclic DG-comodules) and N is an injective DG-comodule, the vector space of morphisms ( M, −→ (0 , N [1])in D ( C – comod ) is isomorphic to the vector space of morphisms M −→ K (cid:3) F N in Hot ( E – comod ). This vector space can be nonzero even when K is acyclic, becausethe cotensor product of an acyclic DG-comodule and an injective DG-comodule over F can have nonzero cohomology. It suffices to take E to be the field k and F to be thecoalgebra dual to the exterior algebra in one variable k [ ε ] /ε , concentrated in degree 0and endowed with zero differential. Analogously, a left DG-contramodule over C isthe same that a pair ( P, Q ) of left DG-contramodules P over E and Q over F togetherwith a closed morphism Cohom E ( K, P ) −→ Q of left DG-contramodules over F . Thevector space of morphisms ( P, −→ (0 , Q [1]) in D ( C – contra ) can be nonzero evenwhen K is acyclic, because the complex Cohom from an acyclic DG-comodule to aprojective DG-contramodule over E can have nonzero cohomology.2.5. Cotor and Coext of the first kind.
Let N be a right DG-comodule and M be a left DG-comodule over a DG-coalgebra C . Consider the cobar bicomplex N ⊗ k M −→ N ⊗ k C ⊗ k M −→ N ⊗ k C ⊗ k C ⊗ k M −→ · · · and construct its totalcomplex by taking infinite products. Let Cotor C,I ( N, M ) denote the cohomology ofthis total complex.Let M be a left DG-comodule and P be a left DG-contramodule over aDG-coalgebra C . Consider the bar bicomplex · · · −→ Hom k ( M ⊗ k C ⊗ k C, P ) −→ om k ( M ⊗ k C, P ) −→ Hom k ( C, P ) and construct its total complex by taking infinitedirect sums. Let Coext IC ( M, P ) denote the cohomology of this total complex.Let k – vect denote the category of vector spaces over k and k – vect gr denote thecategory of graded vector spaces over k . Theorem. (a)
The functor
Cotor
C,I factorizes through the Cartesian product of thederived categories of right and left DG-comodules over C , so there is a well-definedfunctor Cotor
C,I : D ( comod – C ) × D ( C – comod ) −→ k – vect gr . (b) The functor
Coext IC factorizes through the Cartesian product of the derivedcategories of left DG-comodules and left DG-contramodules over C , so there is awell-defined functor Coext IC : D ( C – comod ) × D ( C – contra ) −→ k – vect gr .Proof. This follows [25, I.3–4] from the fact that a complete and cocomplete filteredcomplex is acyclic whenever the associated graded complex is acyclic [14]. (cid:3)
Let N be a right DG-comodule and P be a left DG-contramodule over aDG-coalgebra C . Consider the bar bicomplex · · · −→ N ⊗ k Hom k ( C ⊗ k C, P ) −→ N ⊗ k Hom k ( C, P ) −→ N ⊗ k P and construct its total complex by taking infinitedirect sums. Let Ctrtor C,I ( N, P ) denote the cohomology of this total complex.Let L and M be left DG-comodules over a DG-coalgebra C . Consider the cobarbicomplex Hom k ( L, M ) −→ Hom k ( L, C ⊗ k M ) −→ Hom k ( L, C ⊗ k C ⊗ k M ) −→ · · · and construct its total complex by taking infinite products. Let Ext IC ( L, M ) denotethe cohomology of this total complex.Let P and Q be left DG-contramodules over C . Consider the cobar bicomplexHom k ( P, Q ) −→ Hom k (Hom k ( C, P ) , Q ) −→ Hom k (Hom k ( C ⊗ k C, P ) , Q ) −→ · · · and construct its total complex by taking infinite products. Let Ext C,I ( P, Q ) denotethe cohomology of this total complex.Just as in the above Theorem, the functors Ctrtor
C,I , Ext IC , and Ext C,I factorizethrough the derived categories of DG-comodules and DG-contramodules.All of these constructions can be extended to A ∞ -comodules and A ∞ -contra-modules over A ∞ -coalgebras; see Remark 7.6. Remark.
Another approach to defining derived functors of cotensor product, coho-momorphisms, contratensor product, etc., whose domains would be Cartesian prod-ucts of the derived categories of DG-comodules and DG-contramodules consists inrestricting these functors to the full subcategories of injective DG-comodules andprojective DG-contramodules in the homotopy categories. To obtain versions of de-rived functors Cotor C and Coext C in this way one would have to restrict the functorsof cotensor product and cohomomorphisms to the homotopy categories of injectiveDG-comodules and projective DG-contramodules in both arguments; resolving onlyone of the arguments does not provide a functor factorizable through the derivedcategory in the other argument [48, section 0.2.3]. To construct version of derived unctors Ctrtor C , Ext C , and Ext C , on the other hand, it suffices to resolve just one of the arguments (the second one, the second one, and the first one, respectively).The versions of Ext C and Ext C so obtained coincide with the functors Hom in thederived categories. It looks unlikely that the derived functors defined in the way ofthis Remark should agree with the derived functors defined above in this subsection.3. Coderived and Contraderived Categories of CDG-Modules
CDG-rings and CDG-modules. A CDG-ring (curved differential gradedring) B = ( B, d, h ) is a triple consisting of an associative graded ring B = L i ∈ Z B i ,an odd derivation d : B −→ B of degree 1, and an element h ∈ B satisfying theequations d ( x ) = [ h, x ] for all x ∈ B and d ( h ) = 0. A morphism of CDG-rings f : B −→ A is a pair f = ( f, a ) consisting of a morphism of graded rings f : B −→ A and an element a ∈ A satisfying the equations f ( d B ( x )) = d A ( f ( x )) + [ a, x ] and f ( h B ) = h A + d A ( a ) + a for all x ∈ B , where B = ( B, d B , h B ) and A = ( A, d A , h A ),while the bracket [ y, z ] denotes the supercommutator of y and z . The composition ofmorphisms is defined by the rule ( f, a ) ◦ ( g, b ) = ( f ◦ g, a + f ( b )). Identity morphismsare the morphisms (id , h ∈ B is called the curvature element of a CDG-ring B . The element a ∈ A is called the change-of-connection element of a CDG-ring morphism f .To any DG-ring structure on a graded ring A one can assign a CDG-ring structureon the same graded ring by setting h = 0. This defines a functor from the categoryof DG-rings to the category of CDG-rings. This functor is faithful, but not fullyfaithful, as non-isomorphic DG-rings may become isomorphic as CDG-rings.A left CDG-module ( M, d M ) over a CDG-ring B is a graded left B -module M = L i ∈ Z M i endowed with a derivation d M : M −→ M compatible with the derivation d B of B and such that d M ( x ) = hx for any x ∈ M . A right CDG-module ( N, d N ) overa CDG-ring B is a graded right B -module N = L i ∈ Z N i endowed with a derivation d N : N −→ N compatible with d B and such that d N ( x ) = − xh for any x ∈ N .Notice that there is no natural way to define a left or right CDG-module structureon the free left or right graded module B over a CDG-ring B . At the same time, anyCDG-ring B is naturally a CDG-bimodule over itself, in the obvious sense (see 3.10for the detailed definition).Let f = ( f, a ) : B −→ A be a morphism of CDG-rings and ( M, d M ) be a leftCDG-module over A . Then the left CDG-module R f M over B is defined as the gradedabelian group M with the graded B -module structure obtained by the restrictionof scalars via f , endowed with the differential d ′ M ( x ) = d M ( x ) + ax for x ∈ M .Analogously, let ( N, d N ) be a right CDG-module over A . Then the right CDG-module f N over B is defined as the graded module N over B with the graded modulestructure induced by f endowed with the differental d ′ N ( x ) = d N ( x ) − ( − | x | xa .For any left CDG-modules L and M over B , the complex of homomorphisms Hom B ( L, M ) from L to M over B is constructed using exactly the same formulas asin 1.1. It turns out that these formulas still define a complex in the CDG-module case,as the h -related terms cancel each other. The same applies to the definitions in thesubsequent two paragraphs in 1.1, which all remain applicable in the CDG-modulecase, including the definitions of the complex of homomorphisms between rightCDG-modules, the tensor product complex of a left and a right CDG-module, etc.The cohomology of CDG-rings and CDG-modules is not defined, though, as theirdifferentials may have nonzero squares. A CDG-algebra over a commutative ring k is a graded k -module with a k -linear CDG-ring structure.So left and right CDG-modules over a given CDG-ring B form DG-categories,which we denote, just as the DG-categories of DG-modules, by DG ( B – mod ) and DG ( mod – B ), respectively. All shifts, twists, infinite direct sums, and infinite directproducts exist in the DG-categories of CDG-modules. The corresponding homotopycategories are denoted by Hot ( B – mod ) and Hot ( mod – B ).Notice that there is no obvious way to define derived categories of CDG-modules,since it is not clear what should be meant by an acyclic CDG-module. Moreover,the functors of restriction of scalars R f related to CDG-isomorphisms f betweenDG-rings may well transform acyclic DG-modules to non-acyclic ones.For a CDG-ring B , we will sometimes denote by B the graded ring B consideredwithout its differential and curvature element (or with the zero differential and cur-vature element). For a left CDG-module M and a right CDG-module N over B , wedenote by M and N the corresponding graded modules (or CDG-modules withzero differential) over B .3.2. Some constructions for DG-categories.
The reader will easily recover thedetails of the constructions sketched below.Let DG be a DG-category. Define the DG-category DG ♮ by the following con-struction. An object of DG ♮ is a pair ( Z, t ), where Z in an object of DG and t ∈ Hom − DG ( Z, Z ) is a contracting homotopy with zero square, i. e., d ( t ) = id X and t = 0. Morphisms ( Z ′ , t ′ ) −→ ( Z ′′ , t ′′ ) of degree n in DG ♮ are morphisms f : Z ′ −→ Z ′′ of degree − n in DG such that d ( f ) = 0 in DG . The differential onthe complex of morphisms in DG ♮ is given by the supercommutator with t , i. e., d ♮ ( f ) = t ′′ f − ( − | f | f t ′ .Obviously, all twists of objects by their Maurer–Cartan endomorphisms (see 1.2)exist in the DG-category DG ♮ . Shifts, (finite or infinite) direct sums, or direct prod-ucts exist in DG ♮ whenever they exist in DG . et B = ( B, d, h ) be a CDG-ring. Construct the DG-ring B ∼ = ( B ∼ , ∂ ) as follows.The graded ring B ∼ is obtained by changing the sign of the grading in the ring B [ δ ],which is in turn constructed by adjoining to B an element δ of degree 1 with therelations [ δ, x ] = d ( x ) for x ∈ B and δ = h . The differential ∂ = ∂/∂δ is defined bythe rules ∂ ( δ ) = 1 and ∂ ( x ) = 0 for all x ∈ B . This construction can be extendedto an equivalence between the categories of CDG-rings and acyclic DG-rings [48,section 0.4.4] (here a DG-ring is called acyclic if its cohomology is the zero ring).There is a natural isomorphism of DG-categories DG ( B – mod ) ♮ ≃ DG ( B ∼ – mod ).Let DG be a DG-category with shifts and cones. Denote by ♮ : X X ♮ thefunctor Z ( DG ) −→ Z ( DG ♮ ) assigning to an object X the object cone(id X )[ −
1] withits standard contracting homotopy t . This functor can be extended in a natural wayto a fully faithful functor DG −→ Z ( DG ♮ ), since not necessarily closed morphismsof degree 0 also induce closed morphisms of the cones of identity endomorphismscommuting with the standard contracting homotopies.The functor ♮ has left and right adjoint functors G + , G − : Z ( DG ♮ ) −→ Z ( DG ),which are given by the rules G + ( Z, t ) = Z and G − ( Z, t ) = Z [1], so G + and G − onlydiffer by a shift. Whenever all infinite direct sums (products) exist in the DG-category DG , the functors G + , G − , and ♮ preserve them.In particular, when DG = DG ( B – mod ), the category Z ( DG ♮ ) can be identified withthe category of graded left B -modules in such a way that the functor ♮ becomes thefunctor M M of forgetting the differential. The category DG ( B – mod ) is thenidentified with the full subcategory consisting of all graded B -modules that admita structure of CDG-module over B .For any DG-category DG , objects of the DG-category DG ♮♮ are triples ( W, t, s ),where W is an object of DG and t , s : W −→ W are endomorphisms of degree − t = 0 = s , ts + st = id W , d ( t ) = id W ,and d ( s ) = 0. Assuming that shifts and cones exist in DG , there is a natural fullyfaithful functor DG −→ DG ♮♮ given by the formula W = cone(id X )[ − DG and all images of idempotent endomorphisms exist in Z ( DG ). Indeed, torecover the object X from the object W , it suffices to take the image of the closedidempotent endomorphism ts of the twisted object W ( − s ).In particular, there is a natural equivalence of DG-categories DG ( B – mod ) ≃ DG ( B ∼∼ – mod ). So the DG-category of CDG-modules over an arbitrary CDG-ring isequivalent to the DG-category of DG-modules over a certain acyclic DG-ring.The “almost involution” DG DG ♮ is not defined on the level of homotopycategories. Indeed, if DG is the DG-category of complexes over an additive cate-gory A containing images of its idempotent endomorphisms, then all objects of theDG-category DG ♮ are contractible, while the DG-category DG ♮♮ is again equivalent tothe DG-category of complexes over A . .3. Coderived and contraderived categories.
Let B be a CDG-ring. Then thecategory Z DG ( B – mod ) of left CDG-modules and closed morphisms between themis an abelian category, so one can speak about exact triples of CDG-modules. Wepresume that morphisms constituting an exact triple are closed. An exact triple ofCDG-modules can be also viewed as a finite complex of CDG-modules, so its totalCDG-module can be assigned to it.A left CDG-module L over B is called absolutely acyclic if it belongs to the min-imal thick subcategory of the homotopy category Hot ( B – mod ) containing the totalCDG-modules of exact triples of left CDG-modules over B . The thick subcategoryof absolutely acyclic CDG-modules is denoted by Acycl abs ( B – mod ) ⊂ Hot ( B – mod ).The quotient category D abs ( B – mod ) = Hot ( B – mod ) / Acycl abs ( B – mod ) is called the absolute derived category of left CDG-modules over B .The thick subcategory Acycl abs ( B – mod ) is often too small and some ways of enlarg-ing it turn out to be useful. A left CDG-module over B is called coacyclic if it belongsto the minimal triangulated subcategory of the homotopy category Hot ( B – mod ) con-taining the total CDG-modules of exact triples of left CDG-modules over B andclosed under infinite direct sums. The coacyclic CDG-modules form a thick subcat-egory of the homotopy category, since a triangulated category with infinite directsums contains images of its idempotent endomorphisms [40]. This thick subcategoryis denoted by Acycl co ( B – mod ) ⊂ Hot ( B – mod ). It is the minimal thick subcategory of Hot ( B – mod ) containing Acycl abs ( B – mod ) and closed under infinite direct sums. The coderived category of left CDG-modules over B is defined as the quotient category D co ( B – mod ) = Hot ( B – mod ) / Acycl co ( B – mod ).Analogously, a left CDG-module over B is called contraacyclic if it belongs to theminimal triangulated subcategory of Hot ( B – mod ) containing the total CDG-modulesof exact triples of left CDG-modules over B and closed under infinite products.The thick subcategory formed by all contraacyclic CDG-modules is denoted by Acycl ctr ( B – mod ) ⊂ Hot ( B – mod ). It is the minimal thick subcategory of Hot ( B – mod )containing Acycl abs ( B – mod ) and closed under infinite products. The contrade-rived category of left CDG-modules over B is defined as the quotient category D ctr ( B – mod ) = Hot ( B – mod ) / Acycl ctr ( B – mod ).All the above definitions can be repeated verbatim for right CDG-modules, so thereare thick subcategories Acycl co ( mod – B ), Acycl ctr ( mod – B ), and Acycl abs ( mod – B ) in Hot ( mod – B ) with the corresponding quotient categories D co ( mod – B ), D ctr ( mod – B ),and D abs ( mod – B ). Examples.
When B is a DG-ring, the coderived and contraderived categories ofDG-modules over B still differ from the derived category of DG-modules and betweeneach other, in general. Indeed, they can even all differ when B is simply a ringconsidered as a DG-ring concentrated in degree 0. For example, let Λ = k [ ε ] /ε e the exterior algebra in one variable over a field k . Then there is an infinitein both directions, acyclic, noncontractible complex of free and cofree Λ-modules · · · −→ Λ −→ Λ −→ · · · , where the differentials are given by the action of ε .This complex of Λ-modules is neither coacyclic, nor contraacyclic. Furthermore,let · · · −→ Λ −→ Λ −→ k −→ −→ k −→ Λ −→ Λ −→ · · · be thecomplexes of canonical truncation of the above doubly infinite complex. Then theformer of these two complexes is contraacyclic and the latter is coacyclic, but not theother way. There also exist finite-dimensional DG-modules (over finite-dimensionalDG-algebras over fields) that are acyclic, but neither coacyclic, nor contraacyclic.The simplest example of this kind is that of the DG-algebra with zero differential B = k [ ε ] /ε , where deg ε = 1, and the DG-module M over B constructed as thefree graded B -module with one homogeneous generator m for which d ( m ) = εm . Allthese assertions follow from the results of 3.4–3.5. Remark.
Much more generally, to define the coderived (contraderived) category, itsuffices to have a DG-category DG with shifts, cones, and arbitrary infinite directsums (products), for which the additive category Z ( DG ) is endowed with an exactcategory structure. Examples of such a situation include not only the categories ofCDG-modules, but also, e. g., the category of complexes over an exact category [48].Then one considers the total objects of exact triples in Z ( DG ) as objects of the ho-motopy category H ( DG ) and takes the quotient category of H ( DG ) by the minimaltriangulated subcategory containing all such objects and closed under infinite directsums (products). It may be advisable to require the class of exact triples in Z ( DG )to be closed with respect to infinite direct sums (products) when working with thisconstruction. A deeper notion of an exact DG-category is discussed in Remark 3.5below, where some results provable in this setting are formulated.3.4. Bounded cases.
Let A a DG-ring. Denote by Hot + ( A – mod ) and Hot − ( A – mod )the homotopy categories of DG-modules over A bounded from below and fromabove, respectively. That is, M ∈ Hot + ( A – mod ) iff M i = 0 for all i ≪ M ∈ Hot − ( A – mod ) iff M i = 0 for all i ≫
0. Set
Acycl ± ( A – mod ) = Acycl ( A – mod ) ∩ Hot ± ( A – mod ), and analogously for Acycl co , ± ( A – mod ) and Acycl ctr , ± ( A – mod ).Clearly, the thick subcategories Acycl co ( A – mod ) and Acycl ctr ( A – mod ) are containedin the thick subcategory Acycl ( A – mod ) for any DG-ring A . Theorem 1.
Assume that A i = 0 for all i > . Then (a) Acycl co , + ( A – mod ) = Acycl + ( A – mod ) and Acycl ctr , − ( A – mod ) = Acycl − ( A – mod ) ; (b) the natural functors Hot ± ( A – mod ) / Acycl ± ( A – mod ) −→ D ( A – mod ) are fullyfaithful; (c) the natural functors Hot ± ( A – mod ) / Acycl co , ± ( A – mod ) −→ D co ( A – mod ) and Hot ± ( A – mod ) / Acycl ctr , ± ( A – mod ) −→ D ctr ( A – mod ) are fully faithful; d) the triangulated subcategories Acycl co ( A – mod ) and Acycl ctr ( A – mod ) generatethe triangulated subcategory Acycl ( A – mod ) .Proof. For a DG-module M over A , denote by τ n M the subcomplexes of canonicalfiltration of M considered as a complex of abelian groups. Due to the condition on A ,these are DG-submodules. Notice that the quotient DG-modules τ n +1 M/τ n M arecontractible for any acyclic DG-module M . Let M ∈ Acycl co , + ( A – mod ). Then onehas τ n M = 0 for n small enough, hence τ n M is coacyclic for all n . It remainsto use the exact triple L n τ n M −→ L n τ n M −→ M in order to show that M is coacyclic. The proof of the second assertion of (a) is analogous. To check (b)and (c), it suffices to notice that whenever a DG-module M is acyclic, coacyclic, orcontraacyclic, the DG-modules τ n M and M/τ n M belong to the same class. Thesame observation allows to deduce (d) from (a). (cid:3) Theorem 2.
Assume that A i = 0 for all i < , the ring A is semisimple, and A = 0 . Then (a) Acycl co , − ( A – mod ) = Acycl − ( A – mod ) and Acycl ctr , + ( A – mod ) = Acycl + ( A – mod ) ; (b) the natural functors Hot ± ( A – mod ) / Acycl ± ( A – mod ) −→ D ( A – mod ) are fullyfaithful; (c) the natural functors Hot ± ( A – mod ) / Acycl co , ± ( A – mod ) −→ D co ( A – mod ) and Hot ± ( A – mod ) / Acycl ctr , ± ( A – mod ) −→ D ctr ( A – mod ) are fully faithful; (d) the triangulated subcategories Acycl co ( A – mod ) and Acycl ctr ( A – mod ) generatethe triangulated subcategory Acycl ( A – mod ) .Proof. Analogous to the proof of Theorem 1, with the only change that instead ofthe DG-submodules τ n M one uses the (nonfunctorial) DG-submodules σ > n M ⊂ M ,which are constructed as follows. For any DG-module M over A and an integer n ,choose a complementary A -submodule K ⊂ M n to the submodule im( d n − : M n − → M n ) ⊂ M n . Set ( σ > n M ) i = 0 for i < n , ( σ > n M ) n = K , and ( σ > n M ) i = M i for i > n .Then σ > n M is a DG-submodule of M and the quotient DG-modules σ > n − M/σ > n M are contractible for any acyclic DG-module M over A . (cid:3) Remark.
The assertion (b) of Theorem 2 holds under somewhat weaker assumptions:the condition that A = 0 can be replaced with the condition that d ( A ) = 0. Theproof remains the same, except that the quotient DG-modules σ > n − M/σ > n M nolonger have to be contractible when M is acyclic.3.5. Semiorthogonality.
Let B be a CDG-ring. Denote by Hot ( B – mod inj ) the fullsubcategory of the homotopy category of left CDG-modules over B formed by allthe CDG-modules M for which the graded module M over the graded ring B isinjective. Analogously, denote by Hot ( B – mod proj ) the full subcategory of the homo-topy category Hot ( B – mod ) formed by all the CDG-modules L for which the graded B -module L is projective. heorem. (a) For any CDG-modules L ∈ Acycl co ( B – mod ) and M ∈ Hot ( B – mod inj ) ,the complex Hom B ( L, M ) is acyclic. (b) For any CDG-modules L ∈ Hot ( B – mod proj ) and M ∈ Acycl ctr ( B – mod ) , thecomplex Hom B ( L, M ) is acyclic.Proof. Part (a): since the functor Hom B ( − , M ) transforms shifts, cones, and infinitedirect sums into shifts, shifted cones, and infinite products, it suffices to considerthe case when L is the total CDG-module of an exact triple of CDG-modules 0 −→ ′ K −→ K −→ ′′ K −→
0. Then Hom B ( L, M ) is the total complex of the bicomplexwith three rows 0 −→ Hom B ( ′′ K, M ) −→ Hom B ( K, M ) −→ Hom B ( ′ K, M ) −→ M is an injective graded B -module, this bicomplex is actually a short exactsequence of complexes, hence its total complex is acyclic. The proof of part (b) iscompletely analogous. (cid:3) Remark.
The assertions of Theorem can be extended to the much more general set-ting of exact DG-categories . Namely, let DG be a DG-category with shifts and cones.The category DG is said to be an exact DG-category if an exact category structureis defined on Z ( DG ♮ ) such that the following conditions formulated in terms of thefunctor ♮ are satisfied. Firstly, morphisms in Z ( DG ) whose images are admissiblemonomorphisms in Z ( DG ♮ ) must admit cokernel morphisms in Z ( DG ♮ ) coming frommorphisms in Z ( DG ). Analogously, morphisms in Z ( DG ) whose images are admissi-ble epimorphisms in Z ( DG ♮ ) must admit kernel morphisms in Z ( DG ♮ ) coming frommorphisms in Z ( DG ). Secondly, the natural triples Z −→ G + ( Z ) ♮ −→ Z [ −
1] in Z ( DG ♮ ) must be exact for all objects Z ∈ DG ♮ . In this case the category Z ( DG )itself acquires an exact category structure in which a triple is exact if and only if itsimage is exact in Z ( DG ♮ ); moreover, the functors G + and G − preserve and reflectexactness of triples. Assume further that all infinite direct sums exist in DG andthe class of exact triples in Z ( DG ♮ ) is closed under infinite direct sums. Then thecoderived category of DG is defined as the quotient category of the homotopy cate-gory H ( DG ) by the minimal triangulated subcategory containing the total objects ofexact triples in Z ( DG ) and closed under infinite direct sums; the objects belongingto the latter subcategory are called coacyclic. The complex of homomorphisms fromany coacyclic object to any object of DG whose image is injective with respect tothe exact category Z ( DG ♮ ) is acyclic. Analogously, assume that all infinite productsexist in DG and the class of exact triples in Z ( DG ♮ ) is closed under infinite prod-ucts. Then the contraderived category of DG is defined as the quotient category ofthe homotopy category H ( DG ) by the minimal triangulated subcategory containingthe total objects of exact triples in Z ( DG ) and closed under infinite products; theobjects belonging to the latter subcategory are called contraacyclic. The complex ofhomomorphisms from any object of DG whose image is projective with respect to theexact category Z ( DG ♮ ) into any contraacyclic object is acyclic. .6. Finite homological dimension case.
Let B be a CDG-ring. Assume that thegraded ring B has a finite left homological dimension (i. e., the homological dimen-sion of the category of graded left B -modules is finite). The next Theorem identifiesthe coderived, contraderived, and absolute derived categories of left CDG-modulesover B and describes them in terms of projective and injective resolutions. Theorem. (a)
The three thick subcategories
Acycl co ( B – mod ) , Acycl ctr ( B – mod ) , and Acycl abs ( B – mod ) in the homotopy category Hot ( B – mod ) coincide. (b) The compositions of functors
Hot ( B – mod inj ) −→ Hot ( B – mod ) −→ D abs ( B – mod ) and Hot ( B – mod proj ) −→ Hot ( B – mod ) −→ D abs ( B – mod ) are both equivalences of tri-angulated categories.Proof. We will show that the minimal triangulated subcategory of
Hot ( B – mod ) con-taining the total CDG-modules of exact triples of CDG-modules and the triangulatedsubcategory Hot ( B – mod ) inj form a semiorthogonal decomposition of Hot ( B – mod ), asdo the triangulated subcategory Hot ( B – mod ) proj and the same minimal triangulatedsubcategory. This implies both (b) and the assertion that this minimal triangulatedsubcategory is closed under infinite direct sums and products, which is even strongerthan (a). By Lemma 1.3, it suffices to construct for any CDG-module M over B closed CDG-module morphisms F −→ M −→ J whose cones belong to the men-tioned minimal triangulated subcategory, while the graded B -modules F and J are projective and injective, respectively.Choose a surjection P −→ M onto the graded B -module M from a projectivegraded B -module P . For any graded left B -module L , denote by G + ( L ) theCDG-module over B freely generated by L . The elements of G + ( L ) are formalexpressions of the form x + dy with x , y ∈ L . The action of B and the differential d on G + ( L ) are given by the obvious rules b ( x + dy ) = bx − ( − | b | d ( b ) y + ( − | b | d ( by )for b ∈ B and d ( x + dy ) = hy + dx implied by the equations in the definition ofa CDG-module. There is a bijective correspondence between morphisms of graded B -modules L −→ M and closed morphisms of CDG-modules G + ( L ) −→ M . Thereis also an exact triple of graded B -modules L −→ G + ( L ) −→ L [ − G + ( P ) −→ M ,where the graded B -module G + ( P ) is projective.Let K be the kernel of the surjective morphism G + ( P ) −→ M (taken in theabelian category Z DG ( B – mod ) of CDG-modules and closed morphisms betweenthem). Applying the same procedure to the CDG-module K in place of M , we obtainthe CDG-module G + ( P ), etc. Since the graded left homological dimension of B isfinite, there exists a nonnegative integer d such that the image Z of the morphism G + ( P d ) −→ G + ( P d − ) taken in the abelian category Z DG ( B – mod ) is projective asa graded B -module. Set F to be the total CDG-module of the finite complex ofCDG-modules Z −→ G + ( P d − ) −→ · · · −→ G + ( P ). Then the graded B -module is projective and the cone of the closed morphism F −→ M , being the totalCDG-module of a finite exact complex of CDG-modules, is homotopy equivalent to aCDG-module obtained from the total CDG-modules of exact triples of CDG-modulesusing the operation of cone repeatedly.Analogously, to obtain the desired morphism M −→ J we start with the construc-tion of the CDG-module G − ( L ) cofreely cogenerated by a graded B -module L . Ex-plicitly, G − ( L ) as a graded abelian group consists of all formal expressions of the form d − x + py , where x , y ∈ L and deg d − x = deg x −
1, deg py = deg y . The differentialon G − ( L ) is given by the formulas d ( d − x ) = px and d ( py ) = d − ( hy ). The action of B is given by the formulas b ( d − x ) = ( − | b | d − ( bx ) and b ( px ) = p ( bx ) + d − ( d ( b ) x ).There is a bijective correspondence between morphisms of graded B -modules f : M −→ L and closed morphisms of CDG-modules g : M −→ G − ( L ) which isdescribed by the formula g ( z ) = d − ( f ( dz )) + pf ( z ). There is also a natural closedisomorphism of CDG-modules G − ( L ) ≃ G + ( L )[1].Arguing as above, we construct an exact complex of CDG-modules and closed mor-phisms between them 0 −→ M −→ G − ( I ) −→ G − ( I − ) −→ · · · , where the graded B -modules G − ( I n ) are injective. Since the graded left homological dimension of B is finite, there exists a nonnegative integer d such that the image Z of the mor-phism G − ( I − d +1 ) −→ G − ( I − d ) taken in the category Z DG ( B – mod ) is injective asa graded B -module. It remains to take J to be the total CDG-module of the finitecomplex of CDG-modules G − ( I ) −→ · · · −→ G − ( I − d − ) −→ Z .When B is a CDG-algebra over a field or a DG-ring, there is an alternative proofanalogous to the proof of Theorem 4.4 below. (cid:3) Remark.
The results of Theorem can be generalized to exact DG-categories in thefollowing way. Let DG be an exact DG-category (as defined in Remark 3.5); assumethat the exact category Z ( DG ♮ ) has a finite homological dimension and enoughinjectives. Then the minimal triangulated subcategory of H ( DG ) containing thetotal objects of exact triples in Z ( DG ) and the full triangulated subcategory of H ( DG ) consisting of all the objects whose images are injective in Z ( DG ♮ ) form asemiorthogonal decomposition of H ( DG ). In particular, this minimal triangulatedsubcategory is closed under infinite direct sums. It would be interesting to deduce thelatter conclusion without assuming existence of injectives, but only finite homologicaldimension of the exact category Z ( DG ♮ ) together with existence of infinite directsums in DG and their exactness in Z ( DG ♮ ).So we have D co ( B – mod ) = D ctr ( B – mod ) for any CDG-ring B such that the gradedring B has a finite left homological dimension. There are also other situations whenthe coderived and contraderived categories of CDG-modules over a given CDG-ring,or over two different CDG-rings, are naturally equivalent. Several results in thisdirection can be found in 3.9–3.10 and [35]. or a discussion of the assumptions under which the derived and absolute derivedcategories D ( A – mod ) and D abs ( A – mod ) of DG-modules over a DG-ring A coincide,see Corollaries 6.7 and 6.8.2, Remark 6.10, and subsection 9.4.3.7. Noetherian case.
Let B be a CDG-ring. We will need to consider the followingcondition on the graded ring B :( ∗ ) Any countable direct sum of injective graded left B -modules has a finiteinjective dimension as a graded B -module.The condition ( ∗ ) holds, in particular, when the ring B is graded left Noetherian,i. e., satisfies the ascending chain condition for graded left ideals. Indeed, a direct sumof injective graded left B -modules is an injective graded B -module in this case.Of course, the condition ( ∗ ) also holds when B has a finite graded left homologicaldimension. The next Theorem provides a semiorthogonal decomposition of the ho-motopy category Hot ( B – mod ) and describes the coderived category D co ( B – mod ) interms of injective resolutions. Theorem.
Whenever the condition ( ∗ ) is satisfied, the composition of functors Hot ( B – mod inj ) −→ Hot ( B – mod ) −→ D co ( B – mod ) is an equivalence of triangulatedcategories.Proof. It suffices to construct for any CDG-module M a morphism M −→ J in thehomotopy category of CDG-modules over B such that the graded B -module J isinjective and the cone of the morphism M −→ J is coacyclic. To do so, we startas in the proof of Theorem 3.6, constructing an exact complex of CDG-modules0 −→ M −→ G − ( I ) −→ G − ( I − ) −→ · · · with injective graded B -modules G − ( I − n ) . Let I be the total CDG-module of the complex of CDG-modules G − ( I ) −→ G − ( I − ) −→ · · · formed by taking infinite direct sums. Let us showthat the cone of the closed morphism M −→ I is coacyclic.Indeed, there is a general fact that the total CDG-module of an exact complexof CDG-modules 0 −→ E −→ E − −→ · · · bounded from below formed by takinginfinite direct sums is coacyclic. To prove this, notice that our total CDG-module E is the inductive limit of the total CDG-modules X n of the finite exact complexes ofcanonical truncation 0 −→ E −→ · · · −→ E − n −→ K − n −→
0. So there is an exacttriple of CDG-modules and closed morphisms 0 −→ L n X n −→ L n X n −→ E −→ X n are coacyclic. Now it remains to notice either thatthe total CDG-module of this exact triple is absolutely acyclic by definition, or thatthis exact sequence splits in DG ( B – mod ) , and consequently this total CDG-moduleis even contractible.When the graded ring B is left Noetherian, the class of injective graded B -modules is closed with respect to infinite direct sums, so the graded B -module I is injective and we are done. In general, we need to repeat the same procedure or the second time, using the assumption that the graded B -module I has a finiteinjective dimension and arguing as in the proof of Theorem 3.6. Construct an exactcomplex of CDG-modules 0 −→ I −→ G − ( J ) −→ G − ( J − ) −→ · · · with injectivegraded B -modules G − ( J − n ) . There exists a nonnegative integer d such that theimage K of the morphism G − ( J − d +1 ) −→ G − ( J − d ) taken in the abelian category Z DG ( B – mod ) is injective as a graded B -module. Set J to be the total CDG-moduleof the finite complex of CDG-modules G − ( J ) −→ · · · −→ G − ( J − d +1 ) −→ K ; thenthe cone of the morphism I −→ J is absolutely acyclic and the graded B -module J is injective. The composition of closed morphisms M −→ I −→ J now providesthe desired morphism of CDG-modules.When B is a CDG-algebra over a field or a DG-ring, one can also use a version of theconstruction from the proof of Theorem 4.4 below in place of the above constructionwith the functor G − . (cid:3) Notice the difference between the proofs of the above Theorem and Theorem 1.5.While injective resolutions for the derived category of DG-modules are constructed bytotalizing bicomplexes by taking infinite products along the diagonals, to constructinjective resolutions for the coderived category of CDG-modules one has to totalizeby taking infinite direct sums along the diagonals. There is a similar differencefor projective resolutions: to construct them for the derived category, one has tototalize bicomplexes by taking infinite direct sums (see Theorem 1.4), while for thecontraderived category one has to use infinite products (see Theorem 3.8 below). Cf.the proof of Theorem 4.4. For a discussion of this phenomenon, see 0.1–0.2.
Remark.
Let DG be an exact DG-category in the sense of Remark 3.5. Assumethat all infinite direct sums exist in DG and the class of exact triples in Z ( DG ♮ ) isclosed under infinite direct sums. Whenever there are enough injectives in the exactcategory Z ( DG ♮ ) and countable direct sums of injectives have finite injective dimen-sions in Z ( DG ♮ ), the full triangulated subcategory of the homotopy category H ( DG )consisting of all the objects whose images are injective in Z ( DG ♮ ) is equivalent tothe coderived category of DG .3.8. Coherent case.
Let B be a CDG-ring. Consider the following condition on thegraded ring B :( ∗∗ ) Any countable product of projective graded left B -modules has a finite pro-jective dimension as a graded B -module.In particular, the condition ( ∗∗ ) holds whenever the graded ring B is graded rightcoherent (i. e., all its finitely generated graded right ideals are finitely presented) andall flat graded left B -modules have finite projective dimensions over B . Indeed, aproduct of flat graded left modules over a graded right coherent ring is flat [13]. he condition that all flat graded left modules have finite projective dimensionsis satisfied in many cases. This includes graded left perfect rings, for which all flatgraded left modules are projective [2] (so in particular the condition ( ∗∗ ) holds forgraded right Artinian rings B , i. e., whenever B satisfies the descending chain con-dition for graded right ideals). This also includes [19, Th´eor`eme 7.10] all graded rings B of cardinality not exceeding ℵ n for a finite integer n , and some other importantcases (see [35, section 3] and references therein).The next Theorem provides another semiorthogonal decomposition of the homo-topy category Hot ( B – mod ) and describes the contraderived category D ctr ( B – mod ) interms of projective resolutions. Theorem.
Whenever the condition ( ∗∗ ) is satisfied, the composition of functors Hot ( B – mod proj ) −→ Hot ( B – mod ) −→ D ctr ( B – mod ) is an equivalence of triangulatedcategories.Proof. Completely analogous to the proof of Theorem 3.7. Let us only point out thatone can use, e. g., the Mittag-Leffler condition for the vanishing of the derived functorof projective limit (of abelian groups) in order to show that the total CDG-moduleof an exact complex of CDG-modules · · · −→ E −→ E −→ (cid:3) More generally, denote by
Hot ( B – mod fl ) the full subcategory of the homotopy cat-egory of left CDG-modules over B formed by all the CDG-modules L for which thegraded B -module L is flat. Assume only that the graded ring B is graded rightcoherent, so that the class of flat graded left B -modules is closed under infinite prod-ucts. Then by the same argument and Lemma 1.6 we can conclude that the functor Hot ( B – mod fl ) / Acycl ctr ( B – mod ) ∩ Hot ( B – mod fl ) −→ D ctr ( B – mod ) is an equivalenceof triangulated categories. On the other hand, assume only that all flat graded left B -modules have finite projective dimensions. Then an argument similar to the proofof Theorem 3.6 shows that any contraacyclic CDG-module over B that is flat as agraded B -module is homotopy equivalent to a CDG-module obtained from the totalCDG-modules of exact triples of CDG-modules that are flat as graded B -modulesusing the operation of cone repeatedly. Question.
Let us call a left CDG-module over a CDG-ring B completely acyclic ifit belongs to the minimal triangulated subcategory of Hot ( B – mod ) containing thetotal CDG-modules of exact triples of CDG-modules and closed under both infinitedirect sums and infinite products. The quotient category of Hot ( B – mod ) by thethick subcategory of completely acyclic CDG-modules can be called the complete de-rived category of left CDG-modules over B . It was noticed in [32] that the complexHom B ( L, M ) is acyclic for any completely acyclic left CDG-module M over B andany left CDG-module L over B for which the graded B -module L is projective and nitely generated. Therefore, the minimal triangulated subcategory of Hot ( B – mod )containing all the left CDG-modules over B that are projective and finitely generatedas graded B -modules and closed under infinite direct sums is equivalent to a full tri-angulated subcategory of the complete derived category of left CDG-modules. Underwhat assumptions on a CDG-ring B does this full triangulated subcategory coincidewith the whole complete derived category? Since CDG-modules L over B for whichthe graded B -module L is (projective and) finitely generated are compact objectsof Hot ( B – mod ), this question is equivalent to the following one. Under what assump-tions on a CDG-ring B is every left CDG-module M over B such that the complexHom B ( L, M ) is acyclic for every L as above completely acyclic? CDG-modules L arecompact objects of the complete derived category; the question is whether they areits compact generators. (Cf. [10].) The answer to this question is certainly positivewhen B is a DG-ring and the complete derived category of left DG-modules over B coincides with their derived category. This is so for all DG-rings B satisfying the con-ditions of either Theorem 3.4.1 or Theorem 3.4.2, and also in the cases listed in 9.4,when an even stronger assertion holds. Another situation when the answer is positiveis that of Corollary 6.8.1. Finally, when the graded ring B is left Noetherian andhas a finite left homological dimension, CDG-modules L that are finitely generatedand projective as graded B -modules are compact generators of D abs ( B – mod ) by theresults of subsection 3.11.3.9. Gorenstein case.
Let B be a CDG-ring. Assume that the graded ring B is left Gorenstein, in the sense that the classes of graded left B -modules of finiteinjective dimension and finite projective dimension coincide. In particular, if B isa graded quasi-Frobenius ring, i. e., the classes of projective graded left B -modulesand injective graded left B -modules coincide, then B is left Gorenstein. Of course,any graded ring of finite graded left homological dimension is left Gorenstein. Theorem. (a)
The compositions of functors
Hot ( B – mod inj ) −→ Hot ( B – mod ) −→ D co ( B – mod ) and Hot ( B – mod proj ) −→ Hot ( B – mod ) −→ D ctr ( B – mod ) are both equiv-alences of triangulated categories. (b) There is a natural equivalence of triangulated categories D co ( B – mod ) ≃ D ctr ( B – mod ) .Proof. Part (a) follows from Theorems 3.7–3.8, since any left Gorenstein ring sat-isfies the conditions ( ∗ ) and ( ∗∗ ). Indeed, the projective (injective) dimensionsof modules of finite projective (injective) dimensions over a Gorenstein ring arebounded by a constant, so any infinite direct sum of injective modules has a fi-nite projective dimension and any any infinite product of projective modules hasa finite injective dimension. To prove part (b), denote by Hot ( B – mod fpid ) the omotopy category of left CDG-modules over B whose underlying graded mod-ules have a finite projective (injective) dimension over B , and consider the quo-tient category D abs ( B – mod fpid ) of Hot ( B – mod fpid ) by the minimal thick subcate-gory containing the total CDG-modules of exact triples of CDG-modules over B whose underlying graded modules have a finite projective (injective) dimensionover B . Then the argument from the proof of Theorem 3.6 shows that both com-positions of functors Hot ( B – mod inj ) −→ Hot ( B – mod fpid ) −→ D abs ( B – mod fpid ) and Hot ( B – mod proj ) −→ Hot ( B – mod fpid ) −→ D abs ( B – mod fpid ) are equivalences of triangu-lated categories. Alternatively, both functors D abs ( B – mod fpid ) −→ D co ( B – mod ) and D abs ( B – mod fpid ) −→ D ctr ( B – mod ) are equivalences of triangulated categories. (cid:3) Remark.
Let DG be an exact DG-category such that all infinite direct sums andproducts exist in DG and the class of exact triples in Z ( DG ♮ ) is closed under infi-nite direct sums and products. Assume further that there are enough injectives andprojectives in Z ( DG ♮ ) and the classes of objects of finite injective and projectivedimensions coincide in this exact category. Then full triangulated subcategory of H ( DG ) consisting of all objects whose images are injective in Z ( DG ♮ ) is equiva-lent to the coderived category of DG , the full triangulated subcategory of H ( DG )consisting of all objects whose images are projective in Z ( DG ♮ ) is equivalent to thecontraderived category of DG , and the coderived and contraderived categories of DG are naturally equivalent to each other.Let A be a DG-ring for which the graded ring A is left Gorenstein. Then theequivalence of categories D co ( A – mod ) ≃ D ctr ( A – mod ) makes a commutative diagramwith the localization functors D co ( A – mod ) −→ D ( A – mod ) and D ctr ( A – mod ) −→ D ( A – mod ). Consequently, the localization functor D co ( A – mod ) ≃ D ctr ( A – mod ) −→ D ( A – mod ) has both a left and a right adjoint functors, which are given by the pro-jective and injective (resolutions of) DG-modules in the sense of 1.4–1.5 (cf. [34]).The thick subcategory of D co ( A – mod ) ≃ D ctr ( A – mod ) annihilated by this localizationfunctor can be thought of as the domain of the Tate cohomology functor associatedwith A . More precisely, there is the exact functor from D ( A – mod ) to the subcate-gory annihilated by the localization functor that assigns to an object of D ( A – mod )the cone of the natural morphism between its images under the functors left andright adjoint to the localization functor. The Tate cohomology as a functor of twoarguments in D ( A – mod ) is the composition of this functor with the functor Hom inthe triangulated subcategory annihilated by the localization functor.3.10. Finite-over-Gorenstein case.
Let D and E be CDG-rings. A CDG-bimodule K over D and E is a graded D - E -bimodule endowed with a differential d of degree 1that is compatible with both the differentials of D and E and satisfies the equation d ( x ) = h D x − xh E for all x ∈ K . Given a CDG-bimodule K over D and E and a leftCDG-module L over E , there is a natural structure of left CDG-module over D on he tensor product of graded modules K ⊗ E L . The differential on K ⊗ E L is definedby the usual formula. Analogously, given a CDG-bimodule K over D and E and aleft CDG-module M over D , there is a natural structure of left CDG-module over E on the graded module Hom D ( K, M ). Both the graded left E -module Hom D ( K, M )and the differential on it are defined by the usual formulas. So a CDG-bimodule K defines a pair of adjoint functors K ⊗ E − : Hot ( E – mod ) −→ Hot ( D – mod ) andHom D ( K, − ) : Hot ( D – mod ) −→ Hot ( E – mod ).Let B −→ A ′ and B −→ A ′′ be two morphisms of CDG-rings and C be aCDG-bimodule over A ′ and A ′′ with the following properties. The graded ring A ′ is afinitely generated projective graded left module over the graded ring B . The gradedring A ′′ is a finitely generated projective graded right module over B . The graded A ′ - A ′′ -bimodule C is a finitely generated projective graded left and right B -module.Finally, the adjoint functors C ⊗ A ′′ − and Hom A ′ ( C, − ) induce an equivalence betweenthe categories of graded left A ′′ -modules induced from B and graded left A ′ -modulescoinduced from B , transforming the graded left A ′′ -module A ′′ ⊗ B L into the gradedleft A ′ -module Hom B ( A ′ , L ) and vice versa, for any graded left B -module L .Given a morphism of CDG-modules B −→ A ′ , one can recover the CDG-bimodule C as the module of left B -module homomorphisms A ′ −→ B with its naturalCDG-bimodule structure and the CDG-ring A ′′ as the ring opposite to the ring ofleft A ′ -module endomorphisms of C with its natural CDG-ring structure, assum-ing that C = Hom B ( A ′ , B ) is a finitely generated projective graded left B -module.Analogously, given a morphism of CDG-modules B −→ A ′′ , one can recover theCDG-bimodule C as the module of right B -module homomorphisms A ′′ −→ B andthe CDG-ring A ′ as the ring right A ′′ -module endomorphisms of C , assuming that C = Hom B op ( A ′′ , B ) is a finitely generated projective graded right B -module.Assume that the graded ring B is left Gorenstein. Theorem. (a)
The compositions of functors
Hot ( A ′ – mod inj ) −→ Hot ( A ′ – mod ) −→ D co ( A ′ – mod ) and Hot ( A ′′ – mod proj ) −→ Hot ( A ′′ – mod ) −→ D ctr ( A ′′ – mod ) are bothequivalences of triangulated categories. (b) There is a natural equivalence of triangulated categories D co ( A ′ – mod ) ≃ D ctr ( A ′′ – mod ) .Proof. Given a morphism of graded rings B −→ A ′′ such that A ′′ is a finitelygenerated projective graded right B -module, the graded ring A ′′ satisfies the con-dition ( ∗∗ ) provided that the graded ring B satisfies ( ∗∗ ). Analogously, given amorphism of graded rings B −→ A ′ such that A ′ is a finitely generated projec-tive graded left B -module, the graded ring A ′ satisfies the condition ( ∗ ) providedthat the graded ring B satisfies ( ∗ ). This proves part (a). To prove (b), noticethat the contraderived category D ctr ( A ′′ – mod ) is equivalent to the quotient categoryof the homotopy category of CDG-modules over A ′′ which as graded A ′′ -modules re induced from graded B -modules of finite projective (injective) dimension by itsminimal triangulated subcategory containing the total CDG-modules of exact triplesof CDG-modules over A ′′ which as exact triples of graded A ′′ -modules are inducedfrom exact triples of graded B -modules of finite projective (injective) dimension.The proof of this assertion follows the ideas of the proof of Theorems 3.8 and 3.6.Analogously, the coderived category D co ( A ′ – mod ) is equivalent to the quotient cate-gory of the homotopy category of CDG-modules over A ′ which as graded A ′ -modulesare induced from graded B -modules of finite projective (injective) dimension by itsminimal triangulated subcategory containing the total CDG-modules of exact triplesof CDG-modules over A ′ which as exact triples of graded A ′ -modules are inducedfrom exact triples of graded B -modules of finite projective (injective) dimension.Now the pair of adjoint functors related to the CDG-bimodule C induces an equiva-lence between these triangulated categories. (Cf. [48, sections 5.4–5.5].) (cid:3) Finitely generated CDG-modules.
Let B be a CDG-ring. Assume that thegraded ring B is left Noetherian. Denote by Hot ( B – mod fg ) the homotopy category ofleft CDG-modules over B that are finitely generated as graded B -modules, or equiv-alently, finitely generated as CDG-modules over B . Let Acycl abs ( B – mod fg ) denotethe minimal thick subcategory of Hot ( B – mod fg ) containing the total CDG-modulesof exact triples of finitely generated left CDG-modules over B . The quotient cate-gory D abs ( B – mod fg ) = Hot ( B – mod fg ) / Acycl abs ( B – mod fg ) is called the absolute derivedcategory of finitely generated left CDG-modules over B . Theorem 1.
The natural functor D abs ( B – mod fg ) −→ D co ( B – mod ) is fully faithful.In particular, any object of Acycl co ( B – mod ) homotopy equivalent to an object of Hot ( B – mod fg ) is homotopy equivalent to an object of Acycl abs ( B – mod fg ) .Proof. It suffices to show that any morphism L −→ M in Hot ( B – mod ) betweenobjects L ∈ Hot ( B – mod fg ) and M ∈ Acycl co ( B – mod fg ) factorizes through an objectof Acycl abs ( B – mod fg ). Indeed, any left CDG-module over B that can be obtained fromtotal CDG-modules of exact triples of CDG-modules over B using the operations ofcone and infinite direct sum is a filtered inductive limit of its CDG-submodules thatcan be obtained from the total CDG-modules of exact triples of finitely generatedCDG-modules using the operation of cone. (cid:3) It follows from Theorem 3.7 that the objects of D abs ( B – mod fg ) are compact in D co ( B – mod ). The following result (cf. 5.5 and Question 3.8) is due to Dmitry Arinkinand is reproduced here with his kind permission. Theorem 2.
The objects of D abs ( B – mod fg ) form a set of compact generators of thetriangulated category D co ( B – mod ) . roof. Let J be a left CDG-module over B such that the graded B -module J isinjective. Suppose that the complex Hom B ( L, J ) is acyclic for any finitely generatedleft CDG-module L over B . We have to prove that J is contractible. Apply Zorn’sLemma to the ordered set of all pairs ( M, h ), where M is a CDG-submodule in J and h : M −→ J is a contracting homolopy for the identity embedding M −→ J . Itsuffices to check that whenever M = J there exists a CDG-submodule M ⊂ M ′ ⊂ J , M = M ′ and a contracting homotopy h ′ : M ′ −→ J for the identity embedding M ′ −→ J that agrees with h on M . Let M ′ be any CDG-submodule in J properlycontaining M such that the quotient CDG-module M ′ /M is finitely generated. Since J is injective, the graded B -module morphism h : M −→ J of degree − B -module morphism h ′′ : M ′ −→ J of the same degree. Let ι : M −→ J and ι ′ : M ′ −→ J denote the identity embeddings. The map ι ′ − d ( h ′′ ) isa closed morphism of CDG-modules M ′ −→ J vanishing in the restriction to M , soit induces a closed morphism of CDG-modules f : M ′ /M −→ J . By our assumption,there exists a contracting homotopy c : M ′ /M −→ J for f . Denote the related mapof graded B -modules M ′ −→ J of degree − c . Then h ′ = h ′′ + c : M ′ −→ J is a contracting homotopy for ι ′ extending h . (cid:3) When the graded ring B is left Noetherian and has a finite graded left homologicaldimension, the homotopy category Hot ( B – mod fgp ) of left CDG-modules over B thatare projective and finitely generated as graded B -modules is equivalent to the absolutederived category of finitely generated left CDG-modules D abs ( B – mod fg ) over B . Example.
The following simple example of a Z / A be a commutative ring and w ∈ A be anelement (often required to be a nonzero-divisor). Set B = A , B = 0, d B = 0,and h B = w . Then CDG-modules over ( B, d B , h B ) that are free and finitely gener-ated as Z / A -modules are known classically as “matrix factorizations” [16]and CDG-modules over B that are projective and finitely generated as Z / A -modules are known as “B-branes in the Landau–Ginzburg model” [44]. Accordingto the above, the homotopy category of CDG-modules over B that are projective andfinitely generated as Z / A -modules is equivalent to the absolute derived cat-egory D abs ( B – mod fg ) and compactly generates the coderived category D co ( B – mod )whenever A is a regular Noetherian ring of finite Krull dimension.3.12. Tor and Ext of the second kind.
Let B be a CDG-algebra over a com-mutative ring k . Our goal is to define differential derived functors of the secondkind Tor B,II : D abs ( mod – B ) × D abs ( B – mod ) −−→ k – mod gr Ext
IIB : D abs ( B – mod ) op × D abs ( B – mod ) −−→ k – mod gr nd give a simple categorical interpretation of these definitions in the finite homolog-ical dimension case. First we notice that there are enough projective and injectiveobjects in the category Z DG ( B – mod ), and these objects remain projective and in-jective in the category of graded B -modules. To construct these projectives andinjectives, it suffices to apply the functors G + and G − to projective and injectivegraded B -modules (see the proof of Theorem 3.6).Let N and M be a right and a left CDG-module over B . We will considerCDG-module resolutions · · · −→ Q −→ Q −→ N −→ · · · −→ P −→ P −→ M −→
0, i. e., exact sequences of this form in the abelian categories Z DG ( mod – B )and Z DG ( B – mod ). To any such pair of resolutions we assign the total complex T = Tot ⊓ ( Q • ⊗ B P • ) of the tricomplex Q n ⊗ B P m formed by taking infinite prod-ucts along the diagonal planes. Whenever all the graded right B -modules Q n areflat, the cohomology of the total complex T does not depend of the choice of theresolution P • and vice versa. Indeed, whenever all graded modules Q n are flat, thenatural map Tot ⊓ ( Q • ⊗ B P • ) −→ Tor ⊓ ( Q • ⊗ B M ) is a quasi-isomorphism, since it isa quasi-isomorphism on the quotient complexes by the components of the completedecreasing filtration induced by the canonical filtration of the complex P • .Using existence of projective resolutions in the categories Z DG ( mod – B ) and Z DG ( B – mod ), one can see that the assignment according to which the cohomol-ogy H ( T ) of the total complex T corresponds to a pair ( N, M ) whenever either allthe graded right B -modules Q n , or all the graded left B -modules P n are flat de-fines a functor on the category Z DG ( mod – B ) × Z DG ( B – mod ). It factorizes throughthe Cartesian product of the homotopy categories, defining a triangulated functor oftwo variables H DG ( mod – B ) × H DG ( B – mod ) −→ k – mod gr . The latter functor fac-torizes through the Cartesian product of the absolute derived categories, hence thefunctor which we denote by Tor B,II .Analogously, let L and M be left CDG-modules over B . Consider CDG-moduleresolutions · · · −→ P −→ P −→ L −→ −→ M −→ R −→ R − −→ · · · ,i. e., exact sequences of this form in the category Z DG ( B – mod ). To any suchpair of resolutions we assign the total complex T = Tot ⊕ (Hom B ( P • , R • )) of thetricomplex Hom B ( P n , R m ) formed by taking infinite direct sums along the diago-nal planes. The rule according to which the cohomology H ( T ) corresponds to apair ( L, M ) whenever either all the graded left B -modules P n are projective, orall the graded left B -modules R m are injective defines a functor on the category Z DG ( B – mod ) op × Z DG ( B – mod ). This functor factorizes through the Cartesianproduct of the homotopy categories, defining a triangulated functor of two variables,which in turn factorizes through the Cartesian product of the absolute derived cate-gories. Hence the functor which we denote by Ext IIB .Alternatively, assume that B is a flat graded k -module, and so is one of the gradedmodules N and M . Then one can compute Tor B,II ( N, M ) in terms of a CDG version f the bar-complex of B with coefficients in N and M . Namely, the graded complex · · · −→ N ⊗ k B ⊗ k B ⊗ k M −→ N ⊗ k B ⊗ k M −→ N ⊗ k M is endowed withtwo additional differentials, one coming from the differentials on B , N , and M ,and the other one from the curvature element of B (see the proof of Theorem 4.4,subsections 6.1–6.2, or [46, Section 7 of Chapter 5]). Then the total complex of this“bicomplex with three differentials”, constructed by taking infinite products alongthe diagonals, computes the desired modules Tor of the second kind.To check this, consider the bar-complex of B with coefficients in the complexes ofCDG-modules Q • and P • , construct its total complex by taking infinite products alongthe diagonal hyperplanes, and check that the resulting complex is quasi-isomorphicto both the total complex of Q • ⊗ B P • and the total complex of the bar-complex of B with coefficients in N and M . One proves this using the canonical filtrations ofthe complexes Q • and P • and the bar-complex.Analogously, if B is a projective graded k -module, and either L is a projecitvegraded k -module, or M is an injective graded k -module, then one can computeExt IIB ( L, M ) in terms of the total complex of the CDG version of the cobar-complexof B with coefficients in L and M , constructed by taking infinite direct sums alongthe diagonals.Now assume that the graded ring B has a finite weak homological dimension,i. e., the homological dimension of the functor Tor between graded right and left B -modules is finite. Using Lemma 1.6 and the construction from the proof of The-orem 3.6, one can show that the natural functor Hot ( B – mod fl ) / Acycl abs ( B – mod ) ∩ Hot ( B – mod fl ) −→ D abs ( B – mod ) is an equivalence of triangulated categories. Toprove the analogous assertion for the homotopy category Hot ( mod fl – B ) of rightCDG-modules over B that are flat as graded B -modules, it suffices to pass to theopposite CDG-ring B op = ( B op , d B op , h B op ). The latter coincides with B as a gradedabelian group and has the multiplication, differential, and curvature element definedby the formulas a op b op = ( − | a || b | ( ba ) op , d B op ( b op ) = d B ( b ) op , and h B op = − h op B ,where b op denotes the element of B op corresponding to an element b ∈ B .The tensor product of CDG-modules N ⊗ B M over B is acyclic whenever oneof the CDG-modules N and M is coacyclic and another one is flat as a graded B -module. As in 1.6, it follows that N ⊗ B M is also acyclic whenever one of theCDG-modules N and M is simultaneously coacyclic and flat as a graded B -module.So restricting the functor of tensor product over B to either of the Cartesian products Hot ( mod fl – B ) × Hot ( B – mod ) or Hot ( mod – B ) × Hot ( B – mod fl ), one can construct thederived functor of tensor product of CDG-modules, which is defined on the Cartesianproduct of absolute derived categories and factorizes through the Cartesian productof coderived categories. Thus we get a functor D co ( mod – B ) × D co ( B – mod ) −−→ k – mod gr . his derived functor coincides with the above-defined functor Tor B,II , since one canuse finite resolutions P • and Q • in the construction of the latter functor in the finiteweak homological dimension case.Analogously, whenever the graded ring B has a finite left homological dimension,the functor Hom D abs ( B – mod ) ( L, M ) of homomorphisms in the absolute derived categorycoincides with the above-defined functor Ext
IIB ( L, M ).4.
Coderived Category of CDG-Comodulesand Contraderived Category of CDG-Contramodules
CDG-comodules and CDG-contramodules.
Let k be a fixed ground field.We will consider graded vector spaces V over k endowed with homogeneous endo-morphisms d of degree 1 with not necessarily zero squares. The endomorphism d will be called “the differential”. Given two graded vector spaces V and W with thedifferentials d , the differential on the graded tensor product V ⊗ k W is defined by theusual formula d ( v ⊗ w ) = d ( v ) ⊗ w + ( − | v | v ⊗ d ( w ) and the differential on the gradedvector space of homogeneous homomorphisms Hom k ( V, W ) is defined by the usualformula ( df )( v ) = d ( f ( v )) − ( − | f | f ( dv ). The graded vector space k is endowed withthe zero differential.Using a version of Sweedler’s notation, we will denote symbolically the comultipli-cation in a graded coalgebra C by c c (1) ⊗ c (2) . The coaction in a graded leftcomodule M over C will be denoted by x x ( − ⊗ x (0) , while the coaction in agraded right comodule N over C will be denoted by y y (0) ⊗ y (1) . Here x (0) ∈ M , y (0) ∈ N , and x ( − , y (1) ∈ C . The contraaction map Hom k ( C, P ) −→ P of a gradedleft contramodule P over C will be denoted by π P .The graded dual vector space C ∗ = Hom k ( C, k ) to a graded coalgebra C is a gradedalgebra with the multiplication given by the formula ( φ ∗ ψ )( c ) = φ ( c (2) ) ψ ( c (1) ). Anygraded left comodule M over C has a natural structure of a graded left C ∗ -modulegiven by the rule φ ∗ x = φ ( x ( − ) x (0) , while any graded right comodule N over C hasa natural structure of a graded right C ∗ -module given by y ∗ φ = ( − | φ | φ ( y (1) ) y (0) .In particular, the graded coalgebra C itself is a graded C ∗ -bimodule. Any graded leftcontramodule P over C has a natural structure of a graded left C ∗ -module given by φ ∗ p = π P ( c ( − | φ || p | φ ( c ) p ).A CDG-coalgebra over k is a graded coalgebra C endowed with a homogeneousendomorphism d of degree 1 (with a not necessarily zero square) and a homogeneouslinear function h : C −→ k of degree 2 (that is h vanishes on all the components of C except perhaps C − ) satisfying the following equations. Firstly, the comultiplicationmap C −→ C ⊗ k C and the counit map C −→ k must commute with the differentialson C , k , and C ⊗ k C , where the latter two differentials are given by the above rules. econdly, one must have d ( c ) = h ∗ c − c ∗ h and h ( d ( c )) = 0 for all c ∈ C . Ahomogeneous endomorphism d of degree 1 acting on a graded coalgebra C is calledan odd coderivation of degree 1 if it satisfies the first of these two conditions.A morphism of CDG-coalgebras C −→ D is a pair ( f, a ), where f : C −→ D isa morphism of graded coalgebras and a : C −→ k is a homogeneous linear functionof degree 1 such that the equations d D ( f ( c )) = f ( d C ( c )) + f ( a ∗ c ) − ( − | c | f ( c ∗ a )and h D ( f ( c )) = h C ( c ) + a ( d C ( c )) + a ( c ) hold for all c ∈ C . The composition ofmorphisms is defined by the rule ( g, b ) ◦ ( f, a ) = ( g ◦ f, b ◦ f + a ). Identity morphismsare the morphisms (id , category of CDG-coalgebras is defined.A left CDG-comodule over C is a graded left C -comodule M endowed with ahomogeneous linear endomorphism d of degree 1 (with a not necessarily zero square)satisfying the following equations. Firstly, the coaction map M −→ C ⊗ k M mustcommute with the differentials on M and C ⊗ k M . Secondly, one must have d ( x ) = h ∗ x for all x ∈ M . A right CDG-comodule over C is a graded right C -comodule N endowed with a homogeneous linear endomorphism d of degree 1 such that thecoaction map N −→ N ⊗ k C commutes with the differentials and the equation d ( y ) = − y ∗ h holds. A left CDG-contramodule over C is a graded left C -contramodule P endowed with a homogeneous linear endomorphism d of degree 1 such that thecontraaction map Hom k ( C, P ) −→ P commutes with the differentials on Hom k ( C, P )and P , and the equation d ( p ) = h ∗ p holds. In each of the above three situations,a homogeneous k -linear endomorphism d : M −→ M or d : N −→ N of degree 1 iscalled an odd coderivation of degree 1 compatible with an odd coderivation d : C −→ C of degree 1 or a homogeneous k -linear endomorphism d : P −→ P of degree 1 is calledan odd contraderivation of degree 1 compatible with an odd coderivation d : C −→ C of degree 1 if the first of the two conditions is satisfied.For any morphism of graded coalgebras f : C −→ D there are restriction-of-scalars functors assigning to graded comodules and contramodules over C graded D -comodule and D -contramodule structures on the same graded vector spaces. Nowlet f = ( f, a ) : C −→ D be a morphism of CDG-coalgebras and M be a leftCDG-comodule over C . Then the left CDG-comodule R f M over D is defined byrestricting scalars in the graded C -comodule M via the morphism of graded coalge-bras f : C −→ D and changing the differential d on M by the rule d ′ ( x ) = d ( x )+ a ∗ x .Analogously, let N be a right CDG-comodule over C . Then the right CDG-comodule R f N over D is defined by restricting scalars in the graded comodule N and changingthe differential d on N by the rule d ′ ( y ) = d ( y ) − ( − | y | y ∗ a . Finally, let P be agraded left CDG-contramodule over C . Then the left CDG-contramodule R f P over D is defined by restricting scalars in the graded contramodule P via the morphism f and changing the differential d on P by the rule d ′ ( p ) = d ( p ) + a ∗ p .Whenever N is a right CDG-comodule and M is a left CDG-comodule over aCDG-coalgebra C , the tensor product N (cid:3) C M of the graded comodules N and M over he graded coalgebra C considered as a subspace of the tensor product N ⊗ k M is pre-served by the differential of N ⊗ k M . The restriction of this differential to N (cid:3) C M hasa zero square, which makes N (cid:3) C M a complex. Whenever M is a left CDG-comoduleand P is a left CDG-contramodule over a CDG-coalgebra C , the graded space of co-homomorphisms Cohom C ( M, P ) is an invariant quotient space of the graded spaceHom k ( M, P ) with respect to the differential on Hom k ( M, P ). The induced differentialon Cohom C ( M, P ) has a zero square, which makes Cohom C ( M, P ) a complex.Whenever N is a right CDG-comodule and P is a left CDG-contramodule overa CDG-coalgebra C , the contratensor product N ⊙ C P of the graded comodule N and the graded contramodule P over the graded coalgebra C is an invariant quotientspace of the tensor product N ⊗ k P with respect to the differential on N ⊗ k P . Theinduced differential on N ⊙ C P has a zero square, which makes N ⊙ C P a complex.For any left CDG-comodules L and M over a CDG-coalgebra C , the graded vectorspace of homomorphisms between the graded comodules L and M over the gradedcoalgebra C considered as a subspace of the graded space Hom k ( L, M ) is preservedby the differential on Hom k ( L, M ). The induced differential on Hom C ( L, M ) has azero square, which makes Hom C ( L, M ) a complex. Differentials with zero squares onthe graded vector spaces of homomorphisms Hom C ( R, N ) and Hom C ( P, Q ) for rightCDG-comodules R , N and left CDG-contramodules P , Q over a CDG-coalgebra C are constructed in the analogous way. These constructions define the DG-categories DG ( C – comod ), DG ( comod – C ), and DG ( C – contra ) of left CDG-comodules, rightCDG-comodules, and left CDG-contramodules over C .For a CDG-coalgebra C , we will sometimes denote by C the graded coalgebra C considered without its differential d and linear function h (or with the zero differen-tial and linear function). For left or right CDG-comodules, or CDG-contramodules M , N , or P we will denote by M , N , and P the corresponding graded co-modules and contramodules (or CDG-comodules and CDG-contramodules with zerodifferentials) over C . Notice that for DG being the DG-category of DG-comodulesor DG-contramodules over C , the corresponding additive category Z ( DG ♮ ) can beidentified with the category of graded comodules or contramodules over C ; thefunctor ♮ is identified with the functor of forgetting the differential.All shifts, twists, infinite direct sums, and infinite direct products exist in theDG-categories of CDG-modules and CDG-contramodules. The homotopy categoryof (the DG-category of) left CDG-comodules over C is denoted by Hot ( C – comod ),the homotopy category of right CDG-comodules over C is denoted by Hot ( comod – C ),and the homotopy category of left CDG-contramodules over C is denoted by Hot ( C – contra ). Notice that there is no obvious way to define derived categoriesof CDG-comodules or CDG-contramodules, as there is no notion of cohomol-ogy of a CDG-comodule or a CDG-contramodule, and hence no class of acyclicCDG-comodules or CDG-contramodules. .2. Coderived and contraderived categories.
The absolute derived cate-gories of CDG-comodules and CDG-contramodules, the coderived categories ofCDG-comodules, and the contraderived categories of CDG-contramodules aredefined in the way analogous to that for CDG-modules. These are all particularcases of the general definition sketched in Remarks 3.3 and 3.5. Let us spell outthese definitions in a little more detail.Let C be a CDG-coalgebra. We will consider exact triples in the abelian categories Z DG ( C – comod ) and Z DG ( C – contra ), i. e., exact triples of left CDG-modules orleft CDG-contramodules over C and closed morphisms between them. An exacttriple of CDG-comodules or CDG-contramodules can be viewed as a finite com-plex of CDG-comodules or CDG-contramodules, so the total CDG-comodule orCDG-contramodule is defined for such an exact triple.A left CDG-comodule or left CDG-contramodule over C is called absolutelyacyclic if it belongs to the minimal thick subcategory of the homotopy cat-egory Hot ( C – comod ) or Hot ( C – contra ) containing the total CDG-comodules orCDG-contramodules of exact triples of left CDG-comodules or left CDG-contra-modules over C . The thick subcategories of absolutely acyclic CDG-comodulesand CDG-contramodules are denoted by Acycl abs ( C – comod ) ⊂ Hot ( C – comod )and Acycl abs ( C – contra ) ⊂ Hot ( C – contra ). The quotient categories D abs ( C – comod )and D abs ( C – contra ) of the homotopy categories of left CDG-comodules and leftCDG-contramodules by these thick subcategories are called the absolute derived cat-egories of left CDG-comodules and left CDG-contramodules over C .A left CDG-comodule over C is called coacyclic if it belongs to the minimaltriangulated subcategory of Hot ( C – comod ) containing the total CDG-comodulesof exact triples of left CDG-comodules over C and closed under infinite directsums. The thick subcategory formed by all coacyclic CDG-comodules is de-noted by Acycl co ( C – comod ) ⊂ Hot ( C – comod ). The coderived category of leftCDG-comodules over C is defined as the quotient category D co ( C – comod ) = Hot ( C – comod ) / Acycl co ( C – comod ).A left CDG-contramodule over C is called contraacyclic if it belongs to the minimaltriangulated subcategory of Hot ( C – contra ) containing the total CDG-contramodulesof exact triples of left CDG-contramodules over C and closed under infinite prod-ucts. The thick subcategory formed by all contraacyclic CDG-contramodules is de-noted by Acycl ctr ( C – contra ) ⊂ Hot ( C – contra ). The contraderived category of leftCDG-contramodules over C is defined as the quotient category DG ctr ( C – contra ) = Hot ( C – contra ) / Acycl ctr ( C – contra ).All the above definitions for left CDG-comodules can be repeated verbatimfor right CDG-comodules, so there are thick subcategories Acycl co ( comod – C ) and Acycl abs ( comod – C ) in Hot ( comod – C ) with the corresponding quotient categories D co ( comod – C ) and D abs ( comod – C ). .3. Bounded cases.
Let C be a DG-coalgebra. Denote by Hot + ( C – comod ) and Hot + ( C – contra ) the homotopy categories of DG-comodules and DG-contramodulesover C bounded from below, and denote by Hot − ( C – comod ) and Hot − ( C – contra )the homotopy categories of DG-comodules and DG-contramodules over C boundedfrom above. That is, M ∈ Hot + ( C – comod ) iff M i = 0 for all i ≪ P ∈ Hot − ( C – contra ) iff P i = 0 for all i ≫
0; similarly for
Hot − ( C – comod ) and Hot + ( C – contra ). Set Acycl ± ( C – comod ) = Acycl ( C – comod ) ∩ Hot ± ( C – comod ) andanalogously for Acycl ± ( C – contra ); also set Acycl co , ± ( C – comod ) = Acycl co ( C – comod ) ∩ Hot ± ( C – comod ) and analogously for Acycl ctr , ± ( C – contra ).Clearly, one has Acycl co ( C – comod ) ⊂ Acycl ( C – comod ) and Acycl ctr ( C – contra ) ⊂ Acycl ( C – contra ) for any DG-coalgebra C . Theorem 1.
Assume that C i = 0 for i < . Then (a) Acycl co , + ( C – comod ) = Acycl + ( C – comod ) and Acycl ctr , − ( C – contra ) = Acycl − ( C – contra ) ; (b) the natural functors Hot ± ( C – comod ) / Acycl ± ( C – comod ) −→ D ( C – comod ) and Hot ± ( C – contra ) / Acycl ± ( C – contra ) −→ D ( C – contra ) are fully faithful; (c) the natural functors Hot ± ( C – comod ) / Acycl co , ± ( C – comod ) −→ D co ( C – comod ) and Hot ± ( C – contra ) / Acycl ctr , ± ( C – contra ) −→ D ctr ( C – contra ) are fully faithful.Proof. Analogous to the proof of Theorem 3.4.1. (cid:3)
For an ungraded coalgebra E , the following conditions are equivalent: (i) the cate-gory of left comodules over E is semisimple; (ii) the category of right comodules over E is semisimple; (iii) the category of left contramodules over E is semisimple; (iv) E is the sum of its cosimple subcoalgebras, where a coalgebra is called cosimple if itcontains no nonzero proper subcoalgebras. A coalgebra E satisfying these equivalentconditions is called cosemisimple [48, Appendix A]. Theorem 2.
Assume that C i = 0 for i > , the coalgebra C is cosemisimple, and C − = 0 . Then (a) Acycl co , − ( C – comod ) = Acycl − ( C – comod ) and Acycl ctr , + ( C – contra ) = Acycl + ( C – contra ) ; (b) the natural functors Hot ± ( C – comod ) / Acycl ± ( C – comod ) −→ D ( C – comod ) and Hot ± ( C – contra ) / Acycl ± ( C – contra ) −→ D ( C – contra ) are fully faithful; (c) the natural functors Hot ± ( C – comod ) / Acycl co , ± ( C – comod ) −→ D co ( C – comod ) and Hot ± ( C – contra ) / Acycl ctr , ± ( C – contra ) −→ D ctr ( C – contra ) are fully faithful.Proof. Analogous to the proof of Theorem 3.4.2. (cid:3)
Remark.
The assertion (b) of Theorem 2 still holds after one replaces the conditionthat C − = 0 with the weaker condition that d ( C − ) = 0. See Remark 3.4. .4. Injective and projective resolutions.
Let C be a CDG-coalgebra. Denoteby Hot ( C – comod inj ) the full triangulated subcategory of the homotopy category ofleft CDG-comodules over C consisting of all the CDG-comodules M for which thegraded comodule M over he graded coalgebra C is injective. Analogously, de-note by Hot ( C – contra proj ) the full triangulated subcategory of the homotopy cate-gory of left CDG-contramodules over C consisting of all the CDG-contramodules P for which the graded contramodule P over the graded coalgebra C is projec-tive. The next Theorem provides semiorthogonal decompositions of the homotopycategories Hot ( C – comod ) and Hot ( C – contra ), and describes the coderived category D co ( C – comod ) and the contraderived category D ctr ( C – contra ) in terms of injectiveand projective resolutions, respectively. Theorem. (a)
For any CDG-comodules L ∈ Acycl co ( C – comod ) and M ∈ Hot ( C – comod inj ) , the complex Hom C ( L, M ) is acyclic. (b) For any CDG-contramodules P ∈ Hot ( C – contra proj ) and Q ∈ Acycl ctr ( C – contra ) ,the complex Hom C ( P, Q ) is acyclic. (c) The composition of functors
Hot ( C – comod inj ) −→ Hot ( C – comod ) −→ D co ( C – comod ) is an equivalence of triangulated categories. (d) The composition of functors
Hot ( C – contra proj ) −→ Hot ( C – contra ) −→ D ctr ( C – contra ) is an equivalence of triangulated categories.Proof. Parts (a) and (b) are easy; see the proof of Theorem 3.5. Parts (c) and (d)can be proven in the way analogous to that of Theorems 3.7 and 3.8, or alternativelyin the following way. Let us first consider the case of a DG-coalgebra C . For anyDG-comodule M over C , consider the cobar resolution C ⊗ k M −→ C ⊗ k C ⊗ k M −→· · · . This is a complex of DG-comodules over C and closed morphisms between them;denote by J the total DG-comodule of this complex formed by taking infinite directsums. Then the graded C -comodule J is injective and the cone of the closedmorphism M −→ J is coacyclic. Analogously, for a DG-contramodule P over C oneconsiders the bar resolution · · · −→ Hom k ( C ⊗ k C, P ) −→ Hom k ( C, P ) and forms itstotal DG-contramodule by taking infinite products. In the case of a CDG-coalgebra C , the construction has to be modified as follows. Let M be a left CDG-comoduleover C ; consider the graded left C -comodule J = L ∞ n =0 ( C ⊗ n +1 ⊗ k M )[ − n ], wherethe comodule structure on C ⊗ n +1 ⊗ k M comes from the comodule structure on theleftmost factor C and the shift of the grading introduces the appropriate sign inthe graded comodule structure. The differential on J is described as the sum ofthree components ∂ , d , and δ given by the formulas ∂ ( c ⊗ c ⊗ · · · ⊗ c n ⊗ x ) = µ ( c ) ⊗ c ⊗ · · · ⊗ x − c ⊗ µ ( c ) ⊗ c ⊗ · · · ⊗ x + · · · + ( − n +1 c ⊗ c ⊗ · · · ⊗ c n ⊗ λ ( x ),where µ : C −→ C ⊗ C and λ : M −→ C ⊗ M are the comultiplication and coactionmaps, ( − n d ( c ⊗ c ⊗ · · · ⊗ c n ⊗ x ) = d ( c ) ⊗ c ⊗ · · · ⊗ x + ( − | c | c ⊗ d ( c ) ⊗ c ⊗ · · · ⊗ x + · · · + ( − | c | + ··· + | c n | c ⊗ · · · ⊗ c n ⊗ d ( x ), and δ ( c ⊗ c ⊗ · · · ⊗ c n ⊗ x ) = ( c ) c ⊗ c ⊗ c ⊗· · ·⊗ x − h ( c ) c ⊗ c ⊗ c ⊗ c ⊗· · ·⊗ x + · · · +( − n − h ( c n ) c ⊗· · · c n − ⊗ x .The graded C -comodule J with the differential ∂ + d + δ is a CDG-comodule over C ; itis endowed with a closed morphism of CDG-comodules M −→ J . Denote by τ n J thesubspaces of canonical filtration of the vector space J considered as a complex with thegrading n and the differential ∂ ; then τ n J are CDG-subcomodules of J . The quotientCDG-comodules τ n J/τ n − J (taken in the abelian category of CDG-comodules andclosed morphisms) are contractible CDG-comodules, the contracting homotopy beinggiven by the map inverse to the differential induced by the differential ∂ . The onlyexception is the CDG-comodule τ J , which is isomorphic to M . It follows that thecone of the closed morphism M −→ J is coacyclic. (cid:3) Remark.
The following results, which are particular cases of Remark 3.7, generalizesimultaneously the above Theorem and, to some extent, Theorems 3.7–3.8. A topo-logical graded abelian group (with additive topology) is defined as a graded abeliangroup endowed with a system of graded subgroups closed under finite intersectionsand containing with any subgroup all the larger graded subgroups; graded subgroupsbelonging to the system are called open. To any topological graded abelian groupone can assign an (ungraded) topological abelian group by taking the projective limitof the direct sums of all grading components of the graded quotient groups by opengraded subgroups. A topological graded abelian group with a graded ring structure iscalled a topological graded ring if its multiplication can be extended to a topologicalring structure on the associated ungraded topological abelian group. Let us restrictourselves to separated and complete topological graded rings B where open two-sidedgraded ideals form a base of the topology; these are exactly the graded projectivelimits of discrete graded rings. Let ( B, d, h ) be a CDG-ring structure on B suchthat the differential d is continuous; one can easily check that open two-sided gradeddifferential ideals form a base of the topology of B in this case, so B is a projectivelimit of discrete CDG-rings. First assume that all discrete graded quotient rings of B are left Noetherian. A graded left B -module is called discrete if the annihilator ofevery its homogeneous element is an open left ideal in B . Consider the DG-category DG ( B – mod ) of discrete graded left B -modules with CDG-module structures. The cor-responding coderived category D co ( B – mod ) is defined in the obvious way. The gradedleft B -module of continuous homogeneous abelian group homomorphisms from B intoany (discrete) injective graded abelian group is an injective object in the categoryof discrete graded left B -modules. A discrete graded left B -module M is injectiveif and only if for any open two-sided graded ideal J ⊂ B the annihilator of J in M is an injective graded left B/J -module. It follows that there are enough injectivesin the abelian category of discrete graded left B -modules and the class of injectivesis closed under infinite direct sums. For any discrete graded left B -module M thegraded left B -module G − ( M ) is also discrete. So the category Z ( DG ( B – mod ) ♮ ) canbe identified with the category of discrete graded left B -modules and the result of emark 3.7 applies. Thus the coderived category D co ( B – mod ) is equivalent to thehomotopy category of discrete left CDG-modules over B that are injective as dis-crete graded modules. Now assume that all discrete graded quotient rings of B areright Artinian. A pseudo-compact graded right module [18] (see also [31]) over B is a topological graded right module where open graded submodules form a base ofthe topology and all discrete quotient modules have finite length. A pseudo-compactright CDG-module over B is a pseudo-compact graded right module endowed witha CDG-module structure such that the differential is continuous. By the result ofRemark 3.7, the contraderived category of pseudo-compact right CDG-modules over B is equivalent to the homotopy category of pseudo-compact right CDG-modulesthat are projective as pseudo-compact graded modules. Furthermore, it is not dif-ficult to define the DG-category DG ( B – contra ) of left CDG-contramodules over B (cf. [48, Remark A.3]). The corresponding contraderived category D ctr ( B – contra ) isequivalent to the homotopy category of left CDG-contramodules that are projectiveas graded contramodules. The key step is to show that a graded left contramodule P over B is projective if and only if for any open two-sided graded ideal J ⊂ B themaximal quotient contramodule of P whose B -contramodule structure comes from a B/J -(contra)module structure is a projective
B/J -module. More generally, assumethat B has a countable base of the topology and all discrete graded quotient ringsof B are right Noetherian. Define a contraflat graded left contramodule over B as agraded contramodule P such that for any J ⊂ B as above the maximal quotient con-tramodule of P whose B -contramodule structure comes from a B/J -module structureis a flat
B/J -module. Then the contraderived category D ctr ( B – contra ) is equivalentto the homotopy category of left CDG-contramodules over B that are projective asgraded contramodules provided that all contraflat graded left contramodules over B have finite projective dimensions in the abelian category of graded contramodules.4.5. Finite homological dimension case.
Let E be a graded coalgebra. Then thehomological dimensions of the categories of graded right E -comodules, graded left E -comodules, and graded left E -contramodules coincide, as they coincide with thehomological dimensions of the derived functors of cotensor product and cohomomor-phisms on the abelian categories of comodules and contramodules [48, section 0.2.9].The common value of these three homological dimensions we will call the homologicaldimension of the graded coalgebra E .Let C be a CDG-coalgebra. Assume that the graded coalgebra C has a fi-nite homological dimension. The next Theorem identifies the coderived categoryof C -comodules and the contraderived category of C -contramodules with the corre-sponding absolute derived categories. Theorem. (a)
The two thick subcategories
Acycl co ( C – comod ) and Acycl abs ( C – comod ) in the homotopy category Hot ( C – comod ) coincide. b) The two thick subcategories
Acycl ctr ( C – contra ) and Acycl abs ( C – contra ) in thehomotopy category Hot ( C – contra ) coincide.Proof. The proof is analogous to that of Theorem 3.6 and can be based on eitherappropriate versions of the constructions from the proof of that Theorem or theconstructions from the proof of Theorem 4.4. (cid:3)
Finite-dimensional CDG-comodules.
Let C be a CDG-coalgebra. De-note by Hot ( C – comod fd ) the homotopy category of (totally) finite-dimensionalCDG-comodules over C . Let Acycl abs ( C – comod fd ) denote the minimal thick subcate-gory of Hot ( C – comod fd ) containing the total CDG-comodules of exact triples of finite-dimensional left CDG-comodules over C . The quotient category D abs ( C – comod fd ) = Hot ( C – comod fd ) / Acycl abs ( C – comod fd ) is called the absolute derived category of finite-dimensional left CDG-comodules over C . Theorem.
The natural functor D abs ( C – comod fd ) −→ D co ( C – comod ) is fully faithful.In particular, any object of Acycl co ( C – comod ) that is homotopy equivalent to an objectof Hot ( C – comod fd ) is homotopy equivalent to an object of Acycl abs ( C – comod fd ) .Proof. Analogous to the proof of Theorem 3.11.1. (cid:3)
It is explained in 5.5 that the objects of D abs ( C – comod fd ) are compact generatorsof the triangulated category D co ( C – comod ).4.7. Cotor, Coext, and Ctrtor.
Let us define the derived functorsCotor C : D co ( comod – C ) × D co ( C – comod ) −−→ k – vect gr Coext C : D co ( C – comod ) op × D ctr ( C – contra ) −−→ k – vect gr Ctrtor C : D co ( comod – C ) × D ctr ( C – contra ) −−→ k – vect gr for a CDG-coalgebra C . We denote by Hot ( comod inj – C ) the full subcategory of Hot ( comod – C ) formed by all the right CDG-comodules N over C for which thegraded C -comodule N is injective. To check that the composition of functors Hot ( comod inj – C ) −→ Hot ( comod – C ) −→ D co ( comod – C ), one can pass to the oppo-site CDG-coalgebra C op = ( C op , d op , h op ), which coincides with C as a graded vectorspace and has the comultiplication, differential, and curvature defined by the formu-las ( c op ) (1) ⊗ ( c op ) (2) = ( − | c (1) || c (2) | c op(2) ⊗ c op(1) , d op ( c op ) = d ( c ) op , and h op ( c op ) = − h ( c ),where c op denotes the element of C op corresponding to an element c ∈ C .To define the functor Cotor C , restrict the functor of cotensor product (cid:3) C : Hot ( comod – C ) × Hot ( C – comod ) −→ Hot ( k – vect ) to either of the full subcategories Hot ( comod inj – C ) × Hot ( C – comod ) or Hot ( comod – C ) × Hot ( C – comod inj ). The func-tors so obtained factorize through the localization D co ( comod – C ) × D co ( C – comod )and the two induced derived functors D co ( comod – C ) × D co ( C – comod ) −→ k – vect gr re naturally isomorphic to each other. Indeed, the cotensor product N (cid:3) C M isacyclic whenever one of the CDG-comodules N and M is coacyclic and the other isinjective as a graded comodule. This follows from the fact that the functor of cotensorproduct with an injective graded comodule sends exact triples of graded comodulesto exact triples of graded vector spaces. To construct an isomorphism between thetwo induced derived functors, it suffices to notice that both of them are isomorphicto the derived functor obtained by restricting the functor (cid:3) C to the full subcategory Hot ( comod inj – C ) × Hot ( C – comod inj ).To define the functor Coext C , restrict the functor of cohomomorphisms Cohom C : Hot ( C – comod ) op × Hot ( C – contra ) −→ Hot ( k – vect ) to either of the full subcategories Hot ( C – comod inj ) op × Hot ( C – contra ) or Hot ( C – comod ) op × Hot ( C – contra proj ). The func-tors so obtained factorize through the localization D co ( C – comod ) op × D ctr ( C – contra )and the two induced derived functors D co ( C – comod ) op × D ctr ( C – contra ) −→ k – vect gr are naturally isomorphic. Indeed, the complex of cohomomorphisms Cohom C ( M, P )is acyclic whenever either the CDG-comodule M is coacyclic and the CDG-contra-module P is projective as a graded contramodule, or the CDG-comodule M is injec-tive as a graded comodule and the CDG-contramodule P is contraacyclic.To define the functor Ctrtor C , restrict the functor of contratensor product ⊙ C : Hot ( comod – C ) × Hot ( C – contra ) −→ Hot ( k – vect ) to the full subcategory Hot ( comod – C ) × Hot ( C – contra proj ). The functor so obtained factorizes through thelocalization D co ( comod – C ) × D ctr ( C – contra ), so one obtains the desired derived func-tor. Indeed, the contratensor product N ⊙ C P is acyclic whenever the CDG-comodule N is coacyclic and the CDG-contramodule P is projective as a graded contramod-ule. Notice that one can only obtain the functor Ctrtor as the derived functor in itssecond argument, but apparently not in its first argument, as comodules adjusted tocontratensor product most often do not exist.By Lemma 1.3, one can also compute the functor Ext C = Hom D co ( C – comod ) ofhomomorphisms in the coderived category of left CDG-comodules in terms ofinjective resolutions of the second argument and the functor Ext C = Hom D ctr ( C – contra ) of homomorphisms in the contraderived category of left CDG-contramodulesin terms of projective resolutions of the first argument. Namely, one hasExt C ( L, M ) = H (Hom C ( L, M )) whenever the CDG-comodule M is injectiveas a graded C -comodule and Ext C ( P, Q ) = H (Hom C ( P, Q )) whenever theCDG-contramodule P is projective as a graded C -contramodule.For any right CDG-comodule N over C and any complex of k -vector spaces V the differential on Hom k ( N, V ) defined by the usual formula provides a structureof CDG-contramodule over C on Hom k ( N, V ). The functor Hom k ( − , V ) assignscontraacyclic CDG-contramodules to coacyclic CDG-comodules N and so inducesa functor I Hom k ( − , V ) : D co ( comod – C ) −→ D ctr ( C – contra ) on the level of code-rived and contraderived categories. There are natural isomorphisms of functors f two arguments Hom k (Cotor C ( N, M ) , H ( V )) ≃ Coext C ( M, I Hom k ( N, V )) andHom k (Ctrtor C ( N, P ) , H ( V )) ≃ Ext C ( P, I Hom k ( N, V )), where H ( V ) denotes thecohomology of the complex V .4.8. Restriction and extension of scalars.
Let g : C −→ D be a morphism ofCDG-coalgebras. Then any CDG-comodule or CDG-contramodule over C can bealso considered as a CDG-comodule or CDG-contramodule over D , as explainedin 4.1. This defines the restriction-of-scalars functors R g : Hot ( C – comod ) −→ Hot ( D – comod ) and R g : Hot ( C – contra ) −→ Hot ( D – contra ). The functor R g has aright adjoint functor E g given by the formula E g ( N ) = C (cid:3) D N , while the func-tor R g has a left adjoint functor given by the formula E g ( Q ) = Cohom D ( C, Q ); todefine the differentials on E g ( N ) and E g ( Q ), it is simplest to decompose g into anisomorphism of CDG-coalgebras followed by a morphism of CDG-coalgebras with avanishing linear function a .The functors R g and R g obviously map coacyclic CDG-comodules and contraacyclicCDG-contramodules to CDG-comodules and CDG-contramodules of the same kind,and so induce functors D co ( C – comod ) −→ D co ( D – comod ) and D co ( C – contra ) −→ D co ( D – contra ), which we denote by I R g and I R g . The functor E g has a right derivedfunctor R E g obtained by restricting E g to the full subcategory Hot ( D – comod inj ) ⊂ Hot ( D – comod ) and composing it with the localization functor Hot ( C – comod ) −→ D co ( C – comod ). The functor E g has a left derived functor L E g obtained by restricting E g to the full subcategory Hot ( D – contra proj ) ⊂ Hot ( D – contra ) and composing it withthe localization functor Hot ( C – contra ) −→ D ctr ( C – contra ). The functor R E g is rightadjoint to the functor I R g and the functor L E g is left adjoint to the functor I R g .For any two CDG-coalgebras E and F , a CDG-bicomodule K over E and F is agraded vector space endowed with commuting structures of a graded left E -comoduleand a graded right F -comodule and a differential d compatible with both the differ-entials in E and F and satisfying the equation d ( x ) = h E ∗ x − x ∗ h F for all x ∈ K .Notice that a CDG-bicomodule over E and F has no natural structures of a leftCDG-comodule over E or right CDG-comodule over F , as the equations for d aredifferent for CDG-comodules and CDG-bicomodules.CDG-bicomodules over E and F form a DG-category with morphisms ofCDG-bicomodules being homogeneous linear maps satisfying the compatibilityequations for both the graded left E -comodule and graded right F -comodule struc-tures; the differential on morphisms of CDG-bicomodules is defined by the usualformula. The class of coacyclic CDG-bicomodules over E and F is constructed inthe same way as the class of coacyclic CDG-comodules, i. e., one considers exacttriples of CDG-bicomodules and closed morphisms between them, and generates theminimal triangulated subcategory of the homotopy category of CDG-bicomodulescontaining the total CDG-bicomodules of exact triples of CDG-bicomodules and losed under infinite direct sums. A CDG-bicomodule over E and E is called simplya CDG-bicomodule over E .Assume that a CDG-coalgebra C is endowed with an increasing filtration by gradedvector subspaces F C ⊂ F C ⊂ · · · , C = S n F n C that is compatible with thecomultiplication and the differential, that is µ ( F n C ) ⊂ P p + q = n F p C ⊗ F q C and d ( F n C ) ⊂ F n C , where µ denotes the comultiplication map. Then the associatedquotient object gr F C = L n F n C/F n − C becomes a CDG-coalgebra with the co-multiplication and differential induced by those in C , and the counit and the cur-vature linear function h obtained by restricting the corresponding linear functionson C to F C . In particular, F C is also a CDG-coalgebra; it is simultaneously aCDG-subcoalgebra of both C and gr F C and a quotient CDG-coalgebra of gr F C . Theassociated quotient space gr F C , in addition to a CDG-coalgebra structure, has astructure of CDG-bicomodule over F C .Now suppose that both CDG-coalgebras C and D are endowed with increasingfiltrations F as above and the morphism of CDG-coalgebras g : C −→ D preserves thefiltrations. Moreover, let us assume that the morphism of CDG-coalgebras F C −→ F D induced by g is an isomorphism and the cone of the morphism gr F C −→ gr F D of CDG-bicomodules over F D is a coacyclic CDG-bicomodule. Theorem. (a)
Assume that gr F C and gr F D are injective graded right F D -comodules. Then the adjoint functors I R g and R E g are equivalences oftriangulated categories. (b) Assume that gr F C and gr F D are injective graded left F D -comodules.Then the adjoint functors I R g and L E g are equivalences of triangulated categories.Proof. We will prove (a); the proof of (b) is analogous. Let N be a CDG-comoduleover D such that the graded comodule N over D is injective. We have to checkthat the cone of the morphism of CDG-comodules R g ( E g ( N )) = C (cid:3) D N −→ N isa coacyclic CDG-comodule over D . Introduce an increasing filtration F on N bythe rule F n N = λ − ( F n D ⊗ k N ), where λ : N −→ D ⊗ k N denotes the left coactionmap. The filtration F is compatible with the graded comodule structure and thedifferential on N . The induced filtration F on the cotensor product C (cid:3) D N can beobtained by the same construction applied to the CDG-comodule C (cid:3) D N over C .It suffices to check that the associated quotient object gr F cone( C (cid:3) D N → N ) is acoacyclic CDG-comodule over F D . But this associated quotient object is isomorphicto the cotensor product cone(gr F C → gr F D ) (cid:3) F D F N , so it remains to noticethat the cotensor product of the CDG-bicomodule cone(gr F C → gr F D ) over F D with any left CDG-comodule L over F D is a coacyclic left CDG-comodule over F D in our assumptions. To check the latter, one can choose a morphism from L into a CDG-comodule that is injective as a graded comodule such that the cone ofthat morphism of CDG-comodules is coacyclic. Now let M be a CDG-comodule ver C . We have to check that the cone of the morphism M −→ R E g ( I R g ( M )) inthe coderived category of CDG-comodules over C is trivial. To do so, we will need aninjective resolution of R g ( M ) that is natural enough, so that filtrations of C , D , and M would induce a filtration of the resolution in a way compatible with the passage tothe associated quotient objects. One can use either the construction from the proof ofTheorem 4.4, or a version of the construction from the proof of Theorem 3.7 with thecoaction map in the role of a natural embedding of a graded comodule into an injectivegraded comodule. Computing the object R E g ( I R g ( M )) in terms of such a naturalresolution J of the CDG-comodule R g ( M ) over D , we find out that it suffices to checkthat the cone of the morphism gr F M −→ gr F ( C (cid:3) D J ) is a coacyclic CDG-comoduleover F D . But the cones of the morphisms gr F M −→ gr F J and gr F ( C (cid:3) D J ) −→ gr F J are coacyclic CDG-comodules over F D , the latter one in view of the above argumentapplied to the CDG-comodule N = gr F J over the CDG-coalgebra gr F D endowed witha morphism of CDG-coalgebras gr F C −→ gr F D . (cid:3) Comodule-Contramodule Correspondence
Functors Φ C and Ψ C . Let C be a CDG-coalgebra over a field k . For anyleft CDG-contramodule P over C , let Φ C ( P ) denote the left CDG-comodule over C constructed in the following way. The underlying graded vector space of Φ C ( P )is, by the definition, the contratensor product C ⊙ C P , which is a graded quotientspace of the tensor product C ⊗ k P . The graded left C -comodule structure and thedifferential on Φ C ( P ) are induced by the graded left C -comodule structure and thedifferential on C ⊗ k P . The graded left C -comodule structure on C ⊗ k P comes fromthe graded left C -comodule structure on C , while the differential on C ⊗ k P is givenby the standard rule as the differential on the tensor product of graded vector spaceswith differentials. It is straightforward to check that Φ C ( P ) is a left CDG-comodule.For any left CDG-comodule M over C , let Ψ C ( M ) denote the left CDG-contra-module over C constructed as follows. The underlying graded vector space of Ψ C ( M )is, by the definition, the space of comodule homomorphisms Hom C ( C, M ). Thegraded left contramodule structure and the differential on Ψ C ( P ) are obtained by re-stricting the graded left contramodule structure and the differential on Hom k ( C, M )to this graded vector subspace. The graded left C -contramodule structure onHom k ( C, M ) is induced by the graded right C -comodule structure on C , while thedifferential on Hom k ( C, M ) is given by the standard rule as the differential on thespace of homogeneous linear maps between graded vector spaces with differentials.To a morphism f : L −→ M in the DG-category of left CDG-comodules over C , oneassigns the morphism Φ C ( f ) given by the formula c ⊗ x ( − ) | f || c | c ⊗ f ( x ). To a mor-phism f : P −→ Q in the DG-category of left CDG-contramodules over C , one assigns he morphism Ψ C ( f ) given by the formula g f ◦ g . These rules define DG-functorsΦ C : DG ( C – contra ) −→ DG ( C – comod ) and Ψ C : DG ( C – comod ) −→ DG ( C – contra ).The isomorphism between the complexes of morphisms induced by our standardisomorphism Hom k ( C ⊗ k P, M ) ≃ Hom k ( P, Hom k ( C, M )) makes the DG-functorΦ C left adjoint to the DG-functor Ψ C . So there are induced adjoint functors Hot ( C – contra ) −→ Hot ( C – comod ) and Hot ( C – comod ) −→ Hot ( C – contra ), whichwe also denote by Φ C and Ψ C . Example.
Let C be the graded coalgebra over a field k for which the graded dualalgebra C ∗ = k [ x ] is the algebra of polynomials in one variable x of degree 1. Con-sider the category of vector spaces V over k endowed with a complete and cocom-plete filtration F indexed by the integers, that is · · · ⊂ F i − V ⊂ F i V ⊂ · · · ⊂ V andlim −→ F i V ≃ V ≃ lim ←− V /F i − V . Then the graded vector space with the components F i V has a natural structure of a graded contramodule over C and the graded vector spacewith the components V /F i − V has a natural structure of a graded comodule over C ,for any such filtered vector space V . These constructions define an equivalence be-tween the categories of (complete and cocomplete) filtered vector spaces, projectivegraded C -contramodules, and injective graded C -comodules. The equivalence be-tween the latter two categories assigning the injective graded comodule L i V /F i − V to the projective graded contramodule L i F i V and vice versa is given by the functors P C ⊙ C P and M Hom C ( C, M ). The category of filtered vector spaces hasa natural exact category structure; in fact, all exact triples in this exact categorysplit. It follows from the graded version of Theorem 6.5 below (see also Appendix A)that the derived (or homotopy) category of filtered vector spaces is equivalent tothe derived category of the abelian category of graded modules over the graded ring k [ ε ] /ε with deg ε = 1. The functor of forgetting the filtration, acting from the de-rived category of filtered vector spaces to the derived category of vector spaces, canbe interpreted in terms of the derived category of graded k [ ε ] /ε -modules as the Tatecohomology functor (cf. 3.9). Besides, the functor of passing to the associated gradedspace on the derived category of filtered vector spaces corresponds to the functor offorgetting the action of ε on the derived category of graded k [ ε ] /ε -modules.5.2. Correspondence Theorem.
Restricting the functor Φ C to the full triangu-lated subcategory Hot ( C – contra proj ) ⊂ Hot ( C – contra ) and composing it with thelocalization functor Hot ( C – comod ) −→ D co ( C – comod ), we obtain the left derivedfunctor L Φ C : D ctr ( C – contra ) −→ D co ( C – comod ). Restricting the functor Ψ C to thefull triangulated subcategory Hot ( C – comod inj ) ⊂ Hot ( C – comod ) and composing itwith the localization functor Hot ( C – contra ) −→ D ctr ( C – contra ), we obtain the rightderived functor R Ψ C : D co ( C – comod ) −→ D ctr ( C – contra ). Theorem.
The functors L Φ C and R Ψ C are mutually inverse equivalences betweenthe coderived category D co ( C – comod ) and the contraderived category D ctr ( C – contra ) . roof. One can easily see that the functors Φ C and Ψ C between the homotopy cate-gories of CDG-contramodules and CDG-comodules over C map the full triangulatedsubcategories Hot ( C – contra proj ) and Hot ( C – comod inj ) into each other and their re-strictions to these subcategories are mutually inverse equivalences between them.More precisely, the functors P C ⊙ C P and M Hom C ( C, M ) transform thefree graded contramodule Hom k ( C, V ) into the cofree graded comodule C ⊗ k V andvice versa, for any graded vector space V . (cid:3) In particular, let B = ( B, d, h ) be a CDG-algebra over a field k such that thegraded algebra B is (totally) finite-dimensional. Then there is a natural equivalenceof triangulated categories D co ( B – mod ) ≃ D ctr ( B – mod ). Indeed, let C = B ∗ be thegraded dual vector space to B with the CDG-coalgebra structure defined by theformulas c ( ab ) = c (2) ( a ) c (1) ( b ), d C ( c )( b ) = ( − | c | c ( d ( b )), and h C ( c ) = c ( h ) for a , b ∈ B and c ∈ C . Then the DG-categories of left CDG-modules over B , leftCDG-comodules over C , and left CDG-contramodules over C are all isomorphic, so D co ( B – mod ) = D co ( C – comod ) ≃ D ctr ( C – contra ) = D ctr ( B – mod ). This is a particularcase of Theorem 3.10.5.3. Coext and Ext, Cotor and Ctrtor.
For any left CDG-comodules L and M over a CDG-coalgebra C , there is a natural closed morphism of complexes ofvector spaces Cohom C ( L, Ψ C ( M )) −→ Hom C ( L, M ), which is an isomorphism when-ever either of the graded left C -comodules L and M is injective. For any leftCDG-contramodules P and Q over C , there is a natural morphism of complexes ofvector spaces Cohom C (Φ C ( P ) , Q ) −→ Hom C ( P, Q ), which is an isomorphism when-ever either of the graded left C -contramodules P and Q is projective.For any right CDG-comodule N and left CDG-contramodule P over C , thereis a natural closed morphism of complexes of vector spaces N ⊙ C P −→ N (cid:3) C Φ C ( P ), which is an isomorphism whenever either the graded right C -comodule N is injective, or the graded left C -contramodule P is projective.It follows that there are natural isomorphisms of derived functors of two argumentsExt C ( M, L Φ C ( P )) ≃ Coext C ( M, P ) ≃ Ext C ( R Ψ C ( M ) , P ) and Cotor C ( N, M ) ≃ Ctrtor C ( N, R Ψ C ( M )) for M ∈ D co ( C – comod ), N ∈ D co ( comod – C ), and P ∈ D ctr ( C – contra ). In other words, the comodule-contramodule correspondence trans-forms the functor Coext C into the functors Ext C and Ext C , and also it transformsthe functor Cotor C into the functor Ctrtor C .5.4. Relation with extension of scalars.
Let g : C −→ D be a CDG-coalgebramorphism. For any left CDG-comodule N over D such that the graded D -comodule N is injective, there is a natural closed isomorphism Ψ C ( E g ( N )) ≃ E g (Ψ D ( N )) ofCDG-contramodules over C provided by the isomorphisms Hom C ( C, C (cid:3) D N ) ≃ Hom D ( C, N ) ≃ Cohom D ( C, Hom D ( D, N )) of graded vector spaces. Analogously, or any left CDG-contramodule Q over D such that the graded D -contramodule Q is projective, there is a natural closed isomorphism Φ C ( E g ( Q )) ≃ E g (Φ D ( Q ))of CDG-comodules over C provided by the isomorphisms C ⊙ C Cohom D ( C, Q ) ≃ C ⊙ D Q ≃ C (cid:3) D ( D ⊙ D Q ).Notice also that the functors E g and E g preserve the classes of CDG-comodulesand CDG-contramodules that are injective or projective as graded comodules andcontramodules, while the functors Φ and Ψ map these classes into each other. Thuswe have natural isomorphisms of compositions of derived functors R Ψ C ◦ R E g ≃ L E g ◦ R Ψ D and L Φ C ◦ L E g ≃ R E g ◦ L Φ D ; in other words, the equivalences between thecoderived and contraderived categories of comodules and contramodules transformthe derived functor R E g into the derived functor L E g . Identifying D co ( C – comod )with D ctr ( C – contra ) and D co ( D – comod ) with D ctr ( D – contra ), one can say that thefunctor I R g is left adjoint to the functor R E g = L E g , while the functor I R g is rightadjoint to the same functor R E g = L E g .5.5. Proof of Theorem 2.4.
Our first aim is to show that the triangulated category D co ( C – comod ) ≃ D ctr ( C – contra ) is compactly generated for any CDG-coalgebra C .A triangulated category D where all infinite direct sums exist is said to be compactlygenerated if it contains a set of compact objects C (see Remark 1.8.3 for the definition)such that D coincides with the minimal triangulated subcategory of D containing C and closed under infinite direct sums.We will work with the coderived category D co ( C – comod ). It follows from The-orem 4.4 that any finite-dimensional CDG-comodule over C represents a compactobject in D co ( C – comod ), since the full triangulated subcategory Hot ( C – comod inj ) ⊂ Hot ( C – comod ) is closed under infinite direct sums. Let us check that any CDG-co-module over C up to an isomorphism in D co ( C – comod ) can be obtained from finite-dimensional CDG-comodules by iterated operations of cone and infinite direct sum.(Another proof of this result proceeds along the lines of the proof of Theorem 3.11.2.)A graded coalgebra E is called cosemisimple if its homological dimension (see 4.5)is equal to zero. When the abelian group in which the grading of E takes valueshas no torsion of the order equal to the characteristic of k , a graded coalgebra E is cosemisimple if and only if it is cosemisimple as an ungraded coalgebra. For anygraded coalgebra E there exists a unique maximal cosemisimple graded subcoalge-bra E ss ⊂ E , which coincides with the maximal cosemisimple subcoalgebra of theungraded coalgebra E in the above-mentioned assumption. The quotient coalgebra(without counit) E/E ss is conilpotent , i. e., for any element e ∈ E/E ss the image of e under the iterated comultiplication map E/E ss −→ ( E/E ss ) ⊗ n vanishes for n largeenough. One can easily prove these results, e. g., using the fact that any gradedcoalgebra is the union of its finite-dimensional graded subcoalgebras together withthe graded version of the structure theory of finite-dimensional associative algebras. et E = C ∼ be the graded coalgebra for which the category of CDG-comodulesover C and closed morphisms between them is equivalent to the category of gradedcomodules over E (see [48, sections 0.4.4 and 11.2.2] for an explicit construction).Let F n E ⊂ E be the graded subspace formed by all elements e ∈ E whose imagesvanish in ( E/E ss ) ⊗ n +1 . Then F E = E ss , E = S n F n E , and the filtration F n E is compatible with the coalgebra structure on E . For any graded left comodule M over E , set F n M to be the full preimage of F n E ⊗ k M under the comultiplicationmap M −→ E ⊗ k M . Then the filtrations F on E and M are compatible withthe coaction map; in paricular, F n M are E -subcomodules of M and the quotientcomodules F n M/F n − M are comodules over F E . Since F E is a cosemisimple gradedcoalgebra, the comodules F n M/F n − M are direct sums of irreducible comodules,which are finite-dimensional.Any left CDG-comodule M over C can be viewed as a left graded comodule over E ; the above construction provides a filtration F on M such that F n M are CDG-co-modules over C , the embeddings F n M −→ M are closed morphisms, and the quo-tient CDG-comodules F n M/F n − M taken in the abelian category Z DG ( C – comod )are direct sums of finite-dimensional CDG-comodules over C . It follows that M belongs to the minimal triangulated subcategory of D co ( C – comod ) containing finite-dimensional CDG-comodules and closed under infinite direct sums. So we have proventhat D co ( C – comod ) is compactly generated. The full subcategory formed by finite-dimensional CDG-comodules in D co ( C – comod ) is described in 4.6.Let us point out that in the similar way one can prove that D ctr ( C – contra )is the minimal triangulated subcategory of itself containing finite-dimensionalCDG-contramodules and closed under infinite products. Just as for comodules,the category of CDG-contramodules over C and closed morphisms between themis equivalent to the category of graded contramodules over E . Even though the nat-ural decreasing filtration F n P = im Hom k ( E/F n − E, P ) on a graded contramodule P over E associated with the filtration F of E is not always separated, it is al-ways separated and complete for projective graded contramodules and their gradedsubcontramodules, which is sufficient for the argument to work [48, Appendix A].Now let C be a DG-coalgebra. To prove Theorem 2.4(a), notice that the classof quasi-isomorphisms of DG-comodules in the homotopy category Hot ( C – comod )is locally small [54, 10.4.4–5], hence morphisms between any given two ob-jects in D ( C – comod ) form a set rather than a class. The localization functor D co ( C – comod ) −→ D ( C – comod ) preserves infinite direct sums, since the thicksubcategory of acyclic DG-comodules in D co ( C – comod ) is closed under infinite directsums. So it follows from the Brown representability theorem for the compactlygenerated triangulated category D co ( C – comod ) [42, 33] that the localization functor D co ( C – comod ) −→ D ( C – comod ) has a right adjoint functor. The localization functor Hot ( C – comod ) −→ D co ( C – comod ) has a right adjoint by Theorem 4.4, thus the ocalization functor Hot ( C – comod ) −→ D ( C – comod ) also has a right adjoint functor.Then it remains to use Lemma 1.3.Alternatively, notice that D ( C – comod ) is the quotient category of D co ( C – comod )by the thick subcategory which can be represented as the kernel of the forget-ful functor D co ( C – comod ) −→ D ( k – vect ) or the kernel of the homological functor H : D co ( C – comod ) −→ k – vect gr . Both this forgetful functor and this homologicalfunctor preserve infinite direct sums. It follows that this thick subcategory is well-generated [36, section 7] and therefore the localization functor D co ( C – comod ) −→ D ( C – comod ) has a right adjoint functor.To prove Theorem 2.4(b), one can also notice that the class of quasi-isomorphismsof DG-contramodules is locally small in Hot ( C – contra ), the localization func-tor D ctr ( C – contra ) −→ D ( C – contra ) preserves infinite products, and the covari-ant Brown representability [33] for the compactly generated triangulated category D ctr ( C – contra ) implies existence of a left adjoint functor to the localization functor D ctr ( C – contra ) −→ D ( C – contra ). But the following argument is more illuminating.Consider the object P = Hom k ( C, k ) ∈ D ctr ( C – contra ). Notice that D ( C – contra )is the quotient category of D ctr ( C – contra ) by the thick subcategory of all objects Q such that Hom D ctr ( C – contra ) ( P, Q ) = 0. Consider the minimal triangulated subcat-egory of D ctr ( C – contra ) containing P and closed under infinite direct sums. Thistriangulated category is well-generated and therefore the functor of its embeddinginto D ctr ( C – contra ) has a right adjoint functor. It follows that the localizationfunctor D ctr ( C – contra ) −→ D ( C – contra ) has a left adjoint functor whose image co-incides with the minimal triangulated subcategory of D ctr ( C – contra ) containing P and closed under infinite direct sums. The localization functor Hot ( C – comod ) −→ D ctr ( C – contra ) has a left adjoint functor by Theorem 4.4, thus the localization functor Hot ( C – contra ) −→ D ( C – contra ) also has a left adjoint.In addition to the assertions of Theorem, we have proven that the triangulatedsubcategory Hot ( C – contra ) proj coincides with the minimal triangulated subcategoryof Hot ( C – contra ) containing the DG-contramodule Hom k ( C, k ) and closed under in-finite direct sums. Indeed, this is so in the triangulated category D ctr ( C – contra ) andthe triangulated subcategory Hot ( C – contra proj ) ⊂ Hot ( C – contra ), which is equiv-alent to D ctr ( C – contra ), contains Hom k ( C, k ) and is closed under infinite directsums. We do not know whether the triangulated subcategory
Hot ( C – comod ) inj coincides with the minimal triangulated subcategory of Hot ( C – comod ) contain-ing the DG-comodule C and closed under infinite products; the former subcate-gory certainly contains the latter one. Notice that the DG-comodule C and theDG-contramodule Hom k ( C, k ) correspond to each other under the equivalence ofcategories D co ( C – comod ) ≃ D ctr ( C – contra ). (cid:3) Note added three years later : Assuming some large cardinal axioms, one can indeedshow that
Hot ( C – comod ) inj is the minimal triangulated subcategory in Hot ( C – comod ) ontaining the DG-comodule C and closed under infinite products. Equivalently, thecoderived category D co ( C – comod ) has a semiorthogonal decomposition formed by thefull triangulated subcategory of acyclic DG-comodules and the minimal triangulatedsubcategory containing the DG-comodule C and closed under infinite products.It follows that the derived category D ( C – comod ) can be described as the min-imal triangulated subcategory in D co ( C – comod ) ≃ D ctr ( C – contra ) containing thecanonical object C ∈ D co ( C – comod ), while, as we have seen, the derived category D ( C – contra ) is described as the minimal triangulated subcategory in the same tri-angulated category D co ( C – comod ) ≃ D ctr ( C – contra ) containing the same canonicalobject Hom k ( C, k ) ∈ D ctr ( C – contra ) and closed under infinite direct sums. Noticeagain that the coderived ≃ contraderived category is compactly generated, but thecanonical object C ←→ Hom k ( C, k ) is not compact in it, in general.Indeed, a DG-comodule M over a DG-coalgebra C is acyclic if and only if onehas Hom D co ( C – comod ) ( M, C ) = 0. Hence it suffices to show that the minimal triangu-lated subcategory in D co ( C – comod ) containing the object C and closed under infiniteproducts together with its left orthogonal complement form a semiorthogonal decom-position of D co ( C – comod ). By [58, Theorem 2.4], in the assumption of Vopˇenka’sprinciple [55], every triangulated subcategory closed under infinite products in thehomotopy category of a locally presentable category with a stable model categorystructure has this property. For any CDG-coalgebra C over k , a model categorystructure on a locally Noetherian Grothendieck abelian category with the homotopycategory equivalent to D co ( C – comod ) is provided by Theorem 8.2(a).Moreover, this model category is cofibrantly generated with injective morphismsof finite-dimensional CDG-comodules being the generating cofibrations and injec-tive morphisms of finite-dimensional CDG-comodules with the cokernels belongingto Acycl abs ( C – comod fd ) in the role of generating trivial cofibrations (see 4.6 for the no-tation, and the proof of Theorem 3.11.1 for a relevant argument). In fact, the modelcategory of CDG-comodules over C is even finitely generated by these generating setsof morphisms (see [23] for the background definitions). Indeed, any injective map ofCDG-comodules with a finite-dimensional cokernel is a pushout of an injective mapof finite-dimensional CDG-comodules with the same cokernel. So any cofibration ofCDG-comodules is a transfinite composition of our generating cofibrations and anytrivial cofibration is a transfinite composition of our generating trivial cofibrations.A similar and, essentially, much more general result about the coderived (and alsothe contraderived) model structure on CDG-modules over a CDG-ring being cofi-brantly generated can be found in [57, Proposition 1.3.6] (the proof is based on [59,Theorem 2.13]; cf. Remark 8.3).It would be interesting to know how much can one weaken the set-theoreticalaxioms used in the above argument. The assertion of [56, Theorem 9.5] seems to uggest that supercompact cardinals might suffice. On the other hand, some largecardinals may be necessary [58, 2.6–2.7].I am grateful to Greg Stevenson for directing my attention to the paper [58], andto Carles Casacuberta for his explanations and comments on the results contained inthis and several other recent papers of his, including [56]. I also wish to thank HannoBecker for very interesting correspondence.6. Koszul Duality: Conilpotent and Nonconilpotent Cases
Bar and cobar constructions.
Let B = ( B, d, h ) be a CDG-algebra over afield k . We assume that B is nonzero, i. e., the unit element 1 ∈ B is not equalto 0, and consider k = k · B . Let v : B −→ k be ahomogeneous k -linear retraction of the graded vector space B to its subspace k ; set V = ker v ⊂ B . The direct sum decomposition B = V ⊕ k allows one to split themultiplication map m : V ⊗ k V −→ B , the differential d : V −→ B , and the curvatureelement h ∈ B into the components m = ( m V , m k ), d = ( d V , d k ), and h = ( h V , h k ),where m V : V ⊗ k V −→ V , m k : V ⊗ k V −→ k , d V : V −→ V , d k : V −→ k , h V ∈ V ,and h k ∈ k . Notice that the restrictions of the multiplication map and the differentialto k ⊗ k V , V ⊗ k k , k ⊗ k k , and k are uniquely determined by the axioms of a gradedalgebra and its derivation. One has h k = 0 for the dimension reasons when B is Z -graded, but h k may be nonzero when B is Z / B + = B/k . Let Bar( B ) = L ∞ n =0 B ⊗ n + be the tensor coalgebra generated bythe graded vector space B + . The comultiplication in Bar( B ) is given by the rule b ⊗ · · · ⊗ b n P nj =0 ( b ⊗ · · · ⊗ b j ) ⊗ ( b j +1 ⊗ · · · ⊗ b n ) and the counit is the projectionto the component B ⊗ ≃ k . The coalgebra Bar( B ) is a graded coalgebra with thegrading given by the rule deg( b ⊗ · · · ⊗ b n ) = deg( b ) + · · · + deg( b n ) − n .Odd coderivations of degree 1 on Bar( B ) are determined by their compositionswith the projection of Bar( B ) to the component B ⊗ ≃ B + ; conversely, any linearmap Bar( B ) → B + of degree 2 gives rise to an odd coderivation of degree 1 onBar( B ). Let d Bar be odd coderivation of degree 1 on Bar( B ) whose compositionswith the projection Bar( B ) −→ B + are given by the rules b ⊗ · · · ⊗ b n n > b ⊗ b ( − | b | +1 m V ( b ⊗ b ), b d V ( b ), and 1 h V , where B + isidentified with V and 1 ∈ B ⊗ . Let h Bar : Bar( B ) −→ k be the linear function givenby the formulas h Bar ( b ⊗ · · · ⊗ b n ) = 0 for n > h Bar ( b ⊗ b ) = ( − | b | +1 h k ( b ⊗ b ), h Bar ( b ) = − d k ( b ), and h Bar (1) = h k . Then Bar v ( B ) = (Bar( B ) , d Bar , h
Bar ) is aCDG-coalgebra over k . The CDG-coalgebra Bar v ( B ) is called the bar-construction of a CDG-algebra B endowed with a homogeneous k -linear retraction v : B −→ k .A retraction v : B −→ k is called an augmentation of a CDG-algebra B if( v,
0) : (
B, d, h ) −→ ( k, ,
0) is a morphism of CDG-algebras; equivalently, v is n augmentation if it is a morphism of graded algebras satisfying the equations v ( d ( b )) = 0 and v ( h ) = 0. A k -linear retraction v is an augmentation if and only ifthe CDG-coalgebra Bar v ( B ) is actually a DG-coalgebra, i. e., h Bar = 0.Let C = ( C, d, h ) be a CDG-coalgebra over k . We assume that C is nonzero,i. e., the counit map ε : C −→ k is a nonzero linear function. Let w : k −→ C be a homogeneous k -linear section of the surjective map of graded vector spaces ε ;set W = coker w . The direct sum decomposition C = W ⊕ k allows one to splitthe comultiplication map µ : C −→ W ⊗ k W , the differential d : C −→ W , andthe curvature linear function h : C −→ k into the components µ = ( µ W , µ k ), d =( d W , d k ), and h = ( h W , h k ), where µ W : W −→ W ⊗ k W , µ k ∈ W ⊗ k W , d W : W −→ W , d k ∈ W , h W : W −→ k , and h k ∈ k . Notice that the compositions of thecomultiplication map with the projections C ⊗ k C −→ k ⊗ k W , W ⊗ k k , k ⊗ k k andthe composition of the differential with the projection (counit) C −→ k are uniquelydetemined by the axioms of a graded coalgebra and a differential compatible withthe coalgebra structure. One has h k = 0 for dimension reasons when B is Z -graded,but h k may be nonzero when B is Z / C + = ker ε . Let Cob( C ) = L ∞ n =0 C ⊗ n + be the tensor (free associative) algebra,generated by the graded vector space C + . The multiplication in Cob( C ) is givenby the rule ( c ⊗ · · · ⊗ c j )( c j +1 ⊗ · · · ⊗ c n ) = c ⊗ · · · ⊗ c n and the unit element is1 ∈ k ≃ C ⊗ . The algebra Cob( C ) is a graded algebra with the grading given by therule deg( c ⊗ · · · ⊗ c n ) = deg c + · · · + deg c n + n .Odd derivations of degree 1 on Cob( C ) are determined by their restrictions to thecomponent C + ≃ C ⊗ ⊂ Cob( C ); conversely, any linear map C + −→ Cob( C ) ofdegree 2 gives rise to an odd derivation of degree 1 on Cob( C ). Let d Cob be theodd derivation on Cob( C ) whose restriction to C + is given by the formula d ( c ) =( − | c (1 ,W ) | +1 c (1 ,W ) ⊗ c (2 ,W ) − d W ( c ) + h W ( c ), where C + is identified with W and µ W ( c ) = c (1 ,W ) ⊗ c (2 ,W ) . Let h Cob ∈ Cob( C ) be the element given by the formula h Cob = ( − | µ (1 ,k ) | +1 µ (1 ,k ) ⊗ µ (2 ,k ) − d k + h k , where µ k = µ (1 ,k ) ⊗ µ (2 ,k ) . Then Cob w ( C ) =(Cob( C ) , d Cob , h
Cob ) is a CDG-algebra over k . The CDG-algebra Cob w ( C ) is calledthe cobar-construction of a CDG-coalgebra C endowed with a homogeneous k -linearsection w : k −→ C of the counit map ε : C −→ k .A section w : k −→ C is called a coaugmentation of a CDG-coalgebra C if( w,
0) : ( k, , −→ ( C, d, h ) is a morphism of CDG-coalgebras; equivalently, w isa coaugmentation if it is a morphism of graded coalgebras satisfying the equations d ◦ w = 0 and h ◦ w = 0. A k -linear section w is a coaugmentation if and only if theCDG-algebra Cob w ( C ) is actually a DG-algebra, i. e., h Cob = 0.For any CDG-algebra B with a k -linear retraction v , the k -linear section w : k −→ Bar v ( B ) given by the embedding of k ≃ B ⊗ into Bar( B ) is a coaugmentation of theCDG-coalgebra Bar v ( B ) if and only if h = 0 in B , i. e., B is a DG-algebra. For anyCDG-coalgebra C with a k -linear section w , the k -linear retraction v : Cob w ( C ) −→ k iven by the projection of Cob( C ) onto C ⊗ ≃ k is an augmentation of theCDG-algebra Cob w ( C ) if and only if h = 0 on C , i. e., C is a DG-coalgebra. Soa (co)augmentation on one side of the (co)bar-construction corresponds to the van-ishing of the curvature element on the other side.Given a CDG-algebra B , changing a retraction v : B −→ k to another retrac-tion v ′ : B −→ k given by the formula v ′ ( b ) = v ( b ) + a ( b ) leads to an isomor-phism of CDG-coalgebras (id , a ) : Bar v ( B ) −→ Bar v ′ ( B ), where a : B + −→ k is a linear function of degree 0 identified with the corresponding linear functionBar( B ) −→ B + −→ k of degree 1. Given a CDG-coalgebra C , changing a sec-tion w : k −→ C to another section w ′ : k −→ C given by the rule w ′ (1) = w (1) + a leads to an isomorphism of CDG-algebras (id , a ) : Cob w ′ ( C ) −→ Cob w ( C ), where a ∈ C + is an element of degree 0 identified with the corresponding element ofCob( C ) ⊃ C + of degree 1. To an isomorphism of CDG-coalgebras of the form(id , a ) : ( C, d, h ) −→ ( C, d ′ , h ′ ) one can assign an isomorphism of the correspond-ing cobar-constructions of the form ( f a ,
0) : Cob w ( C, d, h ) −→ Cob w ( C, d ′ , h ′ ) withthe automorphism f a of the graded algebra Cob( C ) given by the rule c c − a ( c )for c ∈ C + . Here a : C −→ k is a linear function of degree 1.Consequently, there is a functor from the category of CDG-coalgebras to thecategory of CDG-algebras assigning to a CDG-coalgebra C its cobar-constructionCob w ( C ). The cobar-construction is also a functor from the category of coaugmentedCDG-coalgebras to the category of DG-algebras, from the category of DG-coalgebrasto the category of augmented CDG-algebras, and from the category of coaugmentedDG-coalgebras to the category of augmented DG-algebras.Furthermore, let us call a morphism of CDG-algebras ( f, a ) : B −→ A strict ifone has a = 0. Then there is a functor from the category of CDG-algebras andstrict morphisms between them to the category of CDG-coalgebras assigning to aCDG-algebra B its bar-construction Bar v ( B ). The bar-construction is also a functorfrom the category of DG-algebras to the category of coaugmented CDG-coalgebras,from the category of augmented CDG-algebras and strict morphisms between themto the category of DG-coalgebras, and from the category of augmented DG-algebrasto the category of coaugmented DG-coalgebras. Remark.
There is no isomorphism of bar-constructions corresponding to an iso-morphism of CDG-algebras that is not strict. The reason is, essentially, that thereexist no morphisms of tensor coalgebras Bar( B ) that do not preserve their coau-mentations k ≃ B ⊗ −→ Bar( B ), while there do exist coderivations of Bar( B ) notcompatible with the coaugmentation. Moreover, for any augmented CDG-algebra B = ( B, d, h ) with h = 0 the DG-coalgebra Bar v ( B ) is acyclic, i. e., its cohomologyis the zero coalgebra. Indeed, consider the dual DG-algebra Bar v ( B ) ∗ . Its sub-algebra of cocycles of degree zero Z (Bar v ( B ) ∗ ) is complete in the adic topologyof its augmentation ideal ker( Z (Bar v ( B ) ∗ ) → k ), while the ideal of coboundaries m d − ⊂ Z (Bar v ( B ) ∗ ) contains elements not belonging to the augmentation ideal.Thus im d − = Z (Bar v ( B ) ∗ ) and the unit element 1 ∈ Bar v ( B ) ∗ is a coboundary.One can show that the DG-coalgebra Bar v ( B ) considered up to DG-coalgebra isomor-phisms carries no information about a coaugmented CDG-algebra B except for thedimensions of its graded components. Furthermore, for any CDG-algebra ( B, d, h )with h = 0 and any left CDG-module M over B , the CDG-comodule Bar v ( B ) ⊗ τ B,v M and the CDG-contramodule Hom τ B,v (Bar v ( B ) , M ) over the CDG-coalgebra Bar v ( B )are contractible, the notation being introduced in 6.2. (See Remark 7.3.)6.2. Twisting cochains.
Let C = ( C, d C , h C ) be a CDG-coalgebra and B =( B, d B , h B ) be a CDG-algebra over the same field k . We introduce a CDG-algebrastructure on the graded vector space of homogeneous homomorphisms Hom k ( C, B )in the following way. The multiplication in Hom k ( C, B ) is given by the formula( f g )( c ) = ( − | g || c (1) | f ( c (1) ) g ( c (2) ). The differential is given by the standard rule d ( f )( c ) = d B ( f ( c )) − ( − | f | f ( d C ( c )). The curvature element is defined by the for-mula h ( c ) = ε ( c ) h B − h C ( c ) ·
1, where 1 is the unit element of B and ε is the counitmap of C . A homogeneous linear map τ : C −→ B of degree 1 is called a twistingcochain [37, 24, 43] if it satisfies the equation τ + dτ + h = 0 with respect to theabove-defined CDG-algebra structure on Hom k ( C, B ).Let C be a CDG-coalgebra and w : k −→ C be a homogeneous k -linear section ofthe counit map ε . Then the composition τ = τ C,w : C −→ Cob( C ) of the homogeneouslinear maps C −→ W ≃ C + ≃ C ⊗ −→ Cob( C ) is a twisting cochain for C andCob w ( C ). Let B be a CDG-algebra and v : C −→ k be a homogeneous k -linearretraction. Then minus the composition Bar v ( B ) −→ B ⊗ ≃ B + ≃ V −→ B is atwisting cochain τ = τ B,v : Bar v ( B ) −→ B for Bar v ( B ) and B .Let τ : C −→ B be a twisting cochain for a CDG-coalgebra C and a CDG-alge-bra B . Then for any left CDG-module M over B there is a natural structure of leftCDG-comodule over C on the tensor product C ⊗ k M . Namely, the coaction of C in C ⊗ k M is induced by the left coaction of C in itself, while the differential on C ⊗ k M isgiven by the formula d ( c ⊗ x ) = d ( c ) ⊗ x + ( − | c | c ⊗ d ( x ) + ( − | c (1) | c (1) ⊗ τ ( c (2) ) x . Wewill denote the tensor product C ⊗ k M with this CDG-comodule structure by C ⊗ τ M .Furthermore, for any left CDG-comodule N over C there is a natural structure ofleft CDG-module over B on the tensor product B ⊗ k N . Namely, the action of B in B ⊗ k N is induced by the left action of B in itself, while the differential on B ⊗ k N isgiven by the formula d ( b ⊗ y ) = d ( b ) ⊗ y +( − | b | b ⊗ d ( y )+( − | b | +1 bτ ( y ( − ) ⊗ y (0) . Wewill denote the tensor product B ⊗ k N with this CDG-module structure by B ⊗ τ N .The correspondences assigning to a CDG-module M over B the CDG-comodule C ⊗ τ M over C and to a CDG-comodule N over C the CDG-module B ⊗ τ N over B can be extended to DG-functors whose action on morphisms is given by the stan-dard formulas f ∗ ( c ⊗ x ) = ( − | f || c | c ⊗ f ∗ ( x ) and g ∗ ( b ⊗ y ) = ( − | g || b | b ⊗ g ∗ ( y ). he DG-functor C ⊗ τ − : DG ( B – mod ) −→ DG ( C – comod ) is right adjoint to theDG-functor B ⊗ τ − : DG ( C – comod ) −→ DG ( B – mod ).Analogously, for any right CDG-module M over B there is a natural structure ofright CDG-comodule over C on the tensor product M ⊗ k C . The coaction of C in M ⊗ k C is induced by the right coaction of C in itself and the differential on M ⊗ k C is given by the formula d ( x ⊗ c ) = d ( x ) ⊗ c + ( − | x | x ⊗ d ( c ) + ( − | x | +1 xτ ( c (1) ) ⊗ c (2) .We will denote the tensor product M ⊗ k C with this CDG-comodule structure by M ⊗ τ C . For any right CDG-comodule N over C there is a natural structure of rightCDG-module over B on the tensor product N ⊗ k B . Namely, the action of B in N ⊗ k B is induced by the right action of B in itself and the differential on N ⊗ k B isgiven by the formula d ( y ⊗ b ) = d ( y ) ⊗ b + ( − | y | y ⊗ d ( b ) + ( − | y (0) | y (0) ⊗ τ ( y (1) ) b . Wewill denote the tensor product N ⊗ k B with this CDG-module structure by N ⊗ τ B .For any left CDG-module P over B there is a natural structure of left CDG-contra-module over C on the graded vector space of homogeneous linear maps Hom k ( C, P ).The contraaction of C in Hom k ( C, P ) is induced by the right coaction of C in it-self as explained in 2.2. The differential on Hom k ( C, P ) is given by the formula d ( f )( c ) = d ( f ( c )) − ( − | f | f ( d ( c )) + ( − | f || c (1) | τ ( c (1) ) f ( c (2) ) for f ∈ Hom k ( C, P ). Wewill denote the graded vector space Hom k ( C, P ) with this CDG-contramodule struc-ture by Hom τ ( C, P ). For any left CDG-contramodule Q over C there is a naturalstructure of left CDG-module over B on the graded vector space of homogeneous lin-ear maps Hom k ( B, Q ). The action of B in Hom k ( B, Q ) is induced by the right actionof B in itself as explained in 1.5 and 1.7. The differential on Hom k ( B, Q ) is givenby the formula d ( f )( b ) = d ( f ( b )) − ( − | f | f ( d ( b )) + π ( c ( − | f | +1+ | c || b | f ( τ ( c ) b )),where π denotes the contraaction map Hom k ( C, Q ) −→ Q . We will denote the gradedvector space Hom k ( B, Q ) with this CDG-module structure by Hom τ ( B, Q ).The correspondences assigning to a CDG-module P over B the CDG-contramoduleHom τ ( C, P ) over C and to a CDG-contramodule Q over C the CDG-moduleHom τ ( B, Q ) over B can be extended to DG-functors whose action on morphismsis given by the standard formula g ∗ ( f ) = g ◦ f for f : C −→ P or f : B −→ Q .The DG-functor Hom τ ( C, − ) : DG ( B – mod ) −→ DG ( C – contra ) is left adjoint to theDG-functor Hom τ ( B, − ) : DG ( C – contra ) −→ DG ( B – mod ).6.3. Duality for bar-construction.
Let A = ( A, d ) be a DG-algebra over afield k . Choose a homogeneous k -linear retraction v : A −→ k and consider thebar-construction C = Bar v ( A ); then C is a coaugmented CDG-coalgebra. Let τ = τ A,v : C −→ A be the natural twisting cochain. Theorem. (a)
The functors C ⊗ τ − : Hot ( A – mod ) −→ Hot ( C – comod ) and A ⊗ τ − : Hot ( C – comod ) −→ Hot ( A – mod ) induce functors D ( A – mod ) −→ D co ( C – comod ) and D co ( C – comod ) −→ D ( A – mod ) , which are mutually inverse equivalences of triangu-lated categories. b) The functors
Hom τ ( C, − ) : Hot ( A – mod ) −→ Hot ( C – contra ) and Hom τ ( A, − ) : Hot ( C – contra ) −→ Hot ( A – mod ) induce functors D ( A – mod ) −→ D ctr ( C – contra ) and D ctr ( C – contra ) −→ D ( A – mod ) , which are mutually inverse equivalences of triangu-lated categories. (c) The above equivalences of triangulated categories D ( A – mod ) ≃ D co ( C – comod ) and D ( A – mod ) ≃ D ctr ( C – contra ) form a commutative diagram with the equivalenceof triangulated categories D co ( C – comod ) ≃ D ctr ( C – contra ) provided by the derivedfunctors L Φ C and R Ψ C of Theorem 5.2.Proof. Part (a): first notice that for any coacyclic CDG-comodule N over C theDG-module A ⊗ τ N over A is contractible. Indeed, whenever N is the totalCDG-module of an exact triple of CDG-modules A ⊗ τ N is the total DG-moduleof an exact triple of DG-modules that splits as an exact triple of graded A -modules.Secondly, let us check that for any acyclic DG-module M over A the CDG-comodule C ⊗ τ M over C is coacyclic. Introduce an increasing filtration F on the coalgebra C = Cob( A ) by the rule F n Cob( A ) = L j n A ⊗ j + . There is an induced filtration on C ⊗ τ M given by the formula F n ( C ⊗ τ M ) = F n C ⊗ k M . This is a filtration byCDG-subcomodules and the quotient CDG-comodules F n ( C ⊗ τ M ) /F n − ( C ⊗ τ M )have trivial C -comodule structures. So these quotient CDG-comodules can be con-sidered simply as complexes of vector spaces, and as such they are isomorphic to thecomplexes A ⊗ n + ⊗ k M . These complexes are acyclic, and hence coacyclic, whenever M is acyclic. So the CDG-comodule C ⊗ τ M is coacyclic. Since it is cofree as a graded C -comodule, it is even contractible. We have shown that there are induced func-tors D ( A – mod ) −→ D co ( C – comod ) and D co ( C – comod ) −→ D ( A – mod ); it remains tocheck that they are mutually inverse equivalences. For any DG-module M over A , theDG-module A ⊗ τ C ⊗ τ M is isomorphic to the total DG-module of the reduced bar-resolution · · · −→ A ⊗ A + ⊗ A + ⊗ M −→ A ⊗ A + ⊗ M −→ A ⊗ M . So the cone of theadjunction morphism A ⊗ τ C ⊗ τ M −→ M is acyclic, since the reduced bar-resolutionremains exact after passing to the cohomology · · · −→ H ( A ) ⊗ H ( A + ) ⊗ H ( M ) −→ H ( A ) ⊗ H ( M ) −→ H ( M ) −→
0, as explained in the proof of Theorem 1.4. Fora CDG-comodule N over C , let K denote the cone of the adjunction morphism N −→ C ⊗ τ A ⊗ τ N . Let us show that the CDG-comodule K is absolutely acyclic.Introduce a finite increasing filtration on the graded C -comodule K by the rules G − K = 0, G − K = N [1], G K = N [1] ⊕ C ⊗ k k ⊗ k N ⊂ N [1] ⊕ C ⊗ k A ⊗ k N ,and G K = K , where C ⊗ k k ⊗ k N is embedded into C ⊗ k A ⊗ k N by the mapinduced by the unit element of A . The differential d on K does not preserve thisfiltration; still one has d ( G i K ) ⊂ G i +1 K . Let ∂ denote the differential induced by d on the associated quotient C -comodule gr G K . Then (gr G K, ∂ ) is an exact complex ofgraded C -comodules; indeed, it is isomorphic to the standard resolution of the gradedcomodule N over the graded tensor coalgebra C . Set L = G − K + d ( G − K ) ⊂ K ;it follows that both L and K/L are contractible CDG-comodules over C . Part (a) is roven; the proof of part (b) is completely analogous (up to duality). To prove (c),it suffices to notice the natural isomorphisms Ψ C ( C ⊗ τ M ) ≃ Hom τ ( C, M ) andΦ C (Hom τ ( C, M )) ≃ C ⊗ τ M for a DG-module M over A . (cid:3) Conilpotent duality for cobar-construction.
A graded coalgebra E with-out counit is called conilpotent (cf. 5.5) if it is the union of the kernels of iteratedcomultiplication maps E −→ E ⊗ n . A graded coalgebra D endowed with a coaug-mentation (morphism of coalgebras) w : k −→ D is called conilpotent if the gradedcoalgebra without counit D/w ( k ) is conilpotent. One can easily see that a conilpotentgraded coalgebra has a unique coaugmentation.For a conilpotent graded coalgebra D set F n D to be the kernel of the composition D −→ D ⊗ n +1 −→ ( D/w ( k )) ⊗ n +1 ; then the increasing filtration F on D is compatiblewith the coalgebra structure. We will call a CDG-coalgebra C = ( C, d, h ) conilpotent if it is conilpotent as a graded coalgebra and coaugmented as a CDG-coalgebra. ADG-coalgebra is conilpotent if it is conilpotent as a CDG-coalgebra. For a conilpotentCDG-coalgebra C , the filtration F defined above is compatible with the CDG-coal-gebra structure, i. e., one has d ( F n C ) ⊂ F n C , and in addition, h ( F C ) = 0.Let C be a conilpotent CDG-coalgebra and w : k −→ C be its coaugmentationmap. Consider the cobar-construction A = Cob w ( C ); then A is a DG-algebra. Let τ = τ C,w : C −→ A be the natural twisting cochain. Theorem.
All the assertions of Theorem 6.3 hold for the DG-algebra A , CDG-co-algebra C , and twisting cochain τ as above in place of A , C , and τ from 6.3.Proof. Just as in the proof of Theorem 6.3 one shows that for any coacyclicCDG-comodule N over C the DG-module A ⊗ τ N over A is contractible. Tocheck that the CDG-comodule C ⊗ τ M is coacyclic (and even contractible) for anyacyclic DG-module M over A , one uses the filtration F on the coalgebra C thatwas constructed above and the induced filtration of the CDG-comodule C ⊗ τ M byits CDG-subcomodules F n ( C ⊗ τ M ) = F n C ⊗ k M . The quotient CDG-comodules F n ( C ⊗ τ M ) /F n − ( C ⊗ τ M ) are simply the complexes F n C/F n − C ⊗ k M with thetrivial C -comodule structures, so they are coacyclic whenever M is acyclic. For anyCDG-comodule N over C , the CDG-comodule C ⊗ τ A ⊗ τ N is isomorphic to the re-duced version of the curved cobar-resolution introduced in the proof of Theorem 4.4.So the same argument with the canonical filtration with respect to the cobar differ-ential ∂ proves that the cone of the adjunction morphism N −→ C ⊗ τ A ⊗ τ N iscoacyclic. For a DG-module M over A , denote by K the cocone of the adjunctionmorphism A ⊗ τ C ⊗ τ M −→ M . We will show that the DG-module K is absolutelyacyclic. Introduce a finite decreasing filtration G on the graded A -module K by therules K/G − K = 0, K/G K = M [ − K/G K = A ⊗ k k ⊗ k M ⊕ M [ − K/G K = K , where A ⊗ k C ⊗ k M maps onto A ⊗ k k ⊗ k M by the map induced bythe counit of C . The differential d on K does not preserve this filtration; still one as d ( G i K ) ⊂ G i − K . Let ∂ denote the differential induced by d on the associatedquotient A -module gr G K . Then (gr G K, ∂ ) is an exact complex of graded A -modules;indeed, it is isomorphic to the standard resolution of the graded module M over thegraded tensor algebra A . Set L = G K + d ( G K ) ⊂ L ; it follows that both L and K/L are contractible DG-modules over A . The proof of part (b) is similar (up toduality), and the proof of (c) is analogous to the proof of Theorem 6.3(c). (cid:3) Acyclic twisting cochains.
Let C be a coaugmented CDG-coalgebra witha coaugmentation w and A be a DG-algebra. Then there is a natural bijectivecorrespondence between morphisms of DG-algebras Cob w ( C ) −→ A and twistingcochains τ : C −→ A such that τ ◦ w = 0. Whenever C is a DG-coalgebra, sothat Cob w ( C ) is an augmented DG-algebra, and A is also an augmented DG-algebrawith an augmentation v , a morphism of DG-algebras Cob w ( C ) −→ A preserves theaugmentations if and only if one has v ◦ τ = 0 for the corresponding twisting cochain τ .Let us assume from now on that C is a conilpotent CDG-coalgebra. Then a twistingcochain τ : C −→ A with τ ◦ w = 0 is said to be acyclic if the corresponding morphismof DG-algebras Cob( C ) −→ A is a quasi-isomorphism. Theorem. (a)
The functors C ⊗ τ − : Hot ( A – mod ) −→ Hot ( C – comod ) and A ⊗ τ − : Hot ( C – comod ) −→ Hot ( A – mod ) induce functors D ( A – mod ) −→ D co ( C – comod ) and D co ( C – comod ) −→ D ( A – mod ) , the former of which is right adjoint to the latter.These functors are mutually inverse equivalences of triangulated categories if andonly if the twisting cochain τ is acyclic. (b) The functors
Hom τ ( C, − ) : Hot ( A – mod ) −→ Hot ( C – contra ) and Hom τ ( A, − ) : Hot ( C – contra ) −→ Hot ( A – mod ) induce functors D ( A – mod ) −→ D ctr ( C – contra ) and D ctr ( C – contra ) −→ D ( A – mod ) , the former of which is left adjoint to the latter. Thesefunctors are mutually inverse equivalences of triangulated categories if and only if thetwisting cochain τ is acyclic. (c) Whenever the twisting cochain τ is acyclic, the above equivalences of triangu-lated categories D ( A – mod ) ≃ D co ( C – comod ) and D ( A – mod ) ≃ D ctr ( C – contra ) form acommutative diagram with the equivalence of triangulated categories D co ( C – comod ) ≃ D ctr ( C – contra ) provided by the derived functors L Φ C and R Ψ C of Theorem 5.2. So in particular the twisting cochain τ = τ A,v of 6.3 is acyclic; the twisting cochain τ = τ C,w of 6.4 is acyclic by the definition.Notice that for any acyclic twisting cochain τ the above equivalences of derivedcategories (of the first and the second kind) transform the trivial CDG-comodule k over C into the free DG-module A over A and the trivial CDG-contramodule k over C into the cofree DG-module Hom k ( A, k ) over A . When C is a DG-coalgebra, A is an augmented DG-algebra with an augmentation v , and one has v ◦ τ = 0,these equivalences of exotic derived categories also transform the trivial DG-module k over A into the cofree DG-comodule C over C and into the free DG-contramodule om k ( C, k ) over C . Here the trivial comodule, contramodule, and module structureson k are defined in terms of the coaugmentation w and augmentation v . Proof.
Part (a): Just as in the proofs of Theorems 6.3 and 6.4 one shows that thefunctor N A ⊗ τ M sends coacyclic CDG-comodules to contractible DG-mod-ules and the functor M C ⊗ τ M sends acyclic DG-modules to contractibleCDG-comodules. In order to see that the induced functors are adjoint it sufficesto recall that adjointness of functors can be expressed in terms of adjunction mor-phisms and equations they satisfy; these morphisms obviously continue to exist andthe equations continue to hold after passing to the induced functors between thequotient categories. To prove that these functors are equivalences of triangulatedcategories if and only if τ is an acyclic twisting cochain, it suffices to apply The-orem 6.4 and Theorem 1.7 for the morphism of DG-algebras f : Cob w ( C ) −→ A .Indeed, there are obvious isomorphisms of functors C ⊗ τ M ≃ C ⊗ τ C,w R f ( M ) fora DG-module M over A and A ⊗ τ N ≃ E f (Cob w ( C ) ⊗ τ C,w N ) for a CDG-comodule N over C . The proof of part (b) is completely analogous and uses the functor E f instead of E f . Notice that for any twisting cochain τ , for any CDG-comodule N over C the DG-module A ⊗ τ N over A is projective and for any CDG-contramodule Q over C the DG-module Hom τ ( A, Q ) over A is injective, as one can prove using eitherthe adjointness of the τ -related functors between the homotopy categories, or thefacts that D co ( C – comod ) is generated by the trivial CDG-comodule k as a triangu-lated category with infinite direct sums and D ctr ( C – contra ) is generated by the trivialCDG-contramodule k as a triangulated category with infinite products (see 5.5 forsome details). The proof of part (c) is identical to the proof of Theorem 6.3(c). (cid:3) Notice that for any acyclic twisting cochain τ : C −→ A and any left CDG-comod-ule N over C the complex A ⊗ τ N computes Cotor C ( k, N ) ≃ Ext C ( k, N ). Indeed,the CDG-comodule C ⊗ τ A ⊗ τ N is isomorphic to N in the coderived category ofCDG-comodules over C . This CDG-comodule is also cofree as a graded C -comoduleand one has A ⊗ τ N ≃ k (cid:3) C ( C ⊗ τ A ⊗ τ N ) ≃ Hom C ( k, C ⊗ τ A ⊗ τ N ). Analogously, forany acyclic twisting cochain τ and any left CDG-contramodule Q over C the complexHom τ ( A, Q ) ≃ Cohom C ( k, Hom τ ( C, Hom τ ( A, Q ))) ≃ k ⊙ C Hom τ ( C, Hom τ ( A, Q ))computes Coext C ( k, Q ) ≃ Ctrtor C ( k, Q ). The DG-algebra A itself, considered as acomplex, computes Cotor C ( k, k ) ≃ Ext C ( k, k ).Now let C be a conilpotent DG-coalgebra, A be an augmented DG-algebra with anaugmentation v , and τ : C −→ A be an acyclic twisting cochain for which v ◦ τ = 0.Then for any left DG-module M over A the complex C ⊗ τ M computes Tor A ( k, M ).Indeed, the DG-module A ⊗ τ C ⊗ τ M is isomorphic to M in the derived categoryof DG-modules over A . This DG-module is also projective, as mentioned in theabove proof, and one has C ⊗ τ M ≃ k ⊗ A ( A ⊗ τ C ⊗ τ M ). Analogously, for any leftDG-module P over A the complex Hom τ ( C, P ) ≃ Hom A ( k, Hom τ ( A, Hom τ ( C, M ))) omputes Ext A ( k, P ). The DG-coalgebra C itself, considered as a complex, computesTor A ( k, k ).It follows from the above Theorem that our definion of the coderived category ofCDG-comodules is equivalent to the definition of Lef`evre-Hasegawa and Keller [37, 30]for a conilpotent CDG-coalgebra C .6.6. Koszul generators.
Let A be a DG-algebra over a field k . Suppose that A isendowed with an increasing filtration by graded subspaces k = F A ⊂ F A ⊂ F A ⊂· · · ⊂ A which is compatible with the multiplication, preserved by the differential,and cocomplete, i. e., A = S n F n A . Let gr F A = L n F n A/F n − A be the associatedquotient algebra; it is a bigraded algebra with a grading i induced by the gradingof A and a nonnegative grading n coming from the filtration F . Assume that thealgebra gr F A is Koszul [46, 47, 48] in its nonnegative grading n .Choose a graded subspace V ⊂ F A complementary to k = F A in F A . Noticethat the filtration F on A is determined by the subspace V ⊂ A , as a Koszul algebrais generated by its component of degree 1. We will call F a Koszul filtration and V a Koszul generating subspace of A . Extend V to a subspace V ⊂ A complementaryto k in A and denote by v : A −→ k the projection of A to k along V .Let C ⊂ L n ( F A/k ) ⊗ n be the Koszul coalgebra quadratic dual to gr F A . Re-call that C is constructed as the direct sum of intersections of the form C = L ∞ n =0 T n − j =1 ( F A/k ) ⊗ j − ⊗ k R ⊗ k ( F A/k ) ⊗ n − j − , where R ⊂ ( F A/k ) ⊗ k ( F A/k ) isthe kernel of the multiplication map ( F A/k ) ⊗ −→ F A/F A . In particular, C = k , C = F A/k , and C = R are the low-degree components of C in the grading n . Wewill consider C as a subcoalgebra of the tensor coalgebra Bar( A ) = L n ( A/k ) ⊗ n andendow C with the total grading inherited from the grading of Bar( A ).One can easily see that the graded subcoalgebra C ⊂ Bar v ( A ) is preserved bythe differential of Bar v ( A ), which makes it a CDG-algebra and a CDG-subcoalgebraof Bar v ( A ). The CDG-algebra structure on C does not depend on the choice of asubspace V ⊂ A , but only on the subspace V ⊂ F A . Define the homogeneneouslinear map τ : C −→ A of degree 1 as minus the composition C −→ C = F A/k ≃ V −→ F A ⊂ A . Clearly, C is a conilpotent CDG-coalgebra with the coaugmentation w : k ≃ C −→ C and τ ◦ w = 0. Theorem.
The map τ is an acyclic twisting cochain.Proof. The element τ ∈ Hom k ( C, A ) satisfies the equation τ + dτ + h = 0, since it isthe image of the twisting cochain τ A,v ∈ Hom k (Bar v ( A ) , A ) under the natural strictsurjective morphism of CDG-algebras Hom k (Bar v ( A ) , A ) −→ Hom k ( C, A ) inducedby the embedding C −→ Bar v ( A ). To check that the morphism of DG-algebrasCob w ( C ) −→ A is a quasi-isomorphism, it suffices to pass to the associated quotientobjects with respect to the increasing filtration F on A and the increasing filtration F n Cob w ( C ) induced by the increasing filtration F on C associated with the grading n .Then it remains to use the fact that the coalgebra C is Koszul [47]. (cid:3) Let A be a DG-algebra with a Koszul generating subspace V and the correspond-ing Koszul filtration F . Consider the CDG-coalgebra C and the twisting cochain τ : C −→ A constructed above. Let M be a left DG-module over A ; suppose that M is endowed with an increasing filtration by graded subspaces F M ⊂ F M ⊂ · · · ⊂ M that is compatible with the filtration on A and the action of A on M , preserved bythe differential on M , and cocomplete, i. e., M = S n F n M . Assume that the asso-ciated quotient module gr F M over the associated quotient algebra gr F A is Koszulin its nonnegative grading n . Define the graded subcomodule N ⊂ C ⊗ τ M as theintersection C ⊗ k F M ∩ C ⊗ k S ⊂ C ⊗ k M , where S ⊂ ( F A/k ) ⊗ F M is the kernelof the action map ( F A/k ) ⊗ k F M −→ F M/F M . This is the Koszul comodulequadratic dual to the Koszul module gr F M over gr F A . The subcomodule N is pre-served by the differential on C ⊗ τ M , so it is a CDG-comodule over C . The naturalmorphism of DG-modules A ⊗ τ N −→ M over A is a quasi-isomorphism, as one canshow in the way analogous to the proof of the above Theorem. A dual result holdsfor DG-modules P over A endowed with a complete decreasing filtration satisfyingthe Koszulity condition and the CDG-contramodules P quadratic dual to them. Example.
Let g be a Lie algebra and A = U g be its universal enveloping algebraconsidered as a DG-algebra concentrated in degree 0. Let F be the standard filtrationon U g and V = g ⊂ U g be the standard generating subspace; they are well-knownto be Koszul. Then A is augmented, so C is a DG-coalgebra; it can be identi-fied with the standard homological complex C ∗ ( g ). The functors M C ⊗ τ M and P Hom τ ( C, P ) are isomorphic to the functors of standard homologicaland cohomological complexes M C ∗ ( g , M ) and P C ∗ ( g , P ) with coeffi-cients in complexes of g -modules M and P . Hence we see that these functors induceequivalences between the derived category of g -modules, the coderived category ofDG-comodules over C ∗ ( g ), and the contraderived category of DG-contramodules over C ∗ ( g ). When g and consequently C ∗ ( g ) are finite-dimensional, DG-comodules andDG-contramodules over C ∗ ( g ) can be identified with DG-modules over the standardcohomological complex C ∗ ( g ), so the functors M C ∗ ( g , M ) and P C ∗ ( g , P )induce equivalences between the derived category of g -modules, the coderived cat-egory of DG-modules over C ∗ ( g ), and the contraderived category of DG-modulesover C ∗ ( g ). These results can be extended to the case of a central extension ofLie algebras 0 −→ k −→ g ′ −→ g −→ k and the envelopingalgebra U ′ g = U g ′ / (1 U g ′ − g ′ ) governing representations of g ′ where the central el-ement 1 ∈ k ⊂ g ′ acts by the identity. Choose a section g −→ g ′ of our centralextension and define the generating subspace V ⊂ g ′ ⊂ U ′ g accordingly; then thecorresponding CDG-coalgebra C coincides with the DG-coalgebra C ∗ ( g ) as a graded oalgebra with a coderivation; the 2-cochain C ( g ) −→ k of the central extension g ′ −→ g is the curvature linear function of C . The derived category of U ′ g -modulesis equivalent to the coderived category of CDG-comodules and the contraderivedcategory of CDG-contramodules over this CDG-coalgebra C . Furthermore, for afinite-dimensional Lie algebra g the bounded derived category of finitely generated U ′ g -modules is equivalent to the absolute derived category of finite-dimensionalCDG-comodules over the (finite-dimensional) CDG-coalgebra C , since the algebra U ′ g is Noetherian and has a finite homological dimension (cf. B.5).6.7. Nonconilpotent duality for cobar-construction.
Let C be a CDG-coalge-bra endowed with a homogeneous k -linear section w : k −→ C of the counitmap ε , and let B be a CDG-algebra. Then there is a natural bijective correspon-dence between morphisms of CDG-algebras Cob w ( C ) −→ B and twisting cochains τ : C −→ B . A morphism of CDG-algebras Cob w ( C ) −→ B is strict if and onlyif one has τ ◦ w = 0 for the corresponding twisting cochain τ . Whenever C isa DG-coalgebra, so that Cob w ( C ) is an augmented CDG-algebra, and B is alsoan augmented CDG-algebra with an augmentation v , a morphism of CDG-algebrasCob w ( C ) −→ B preserves the augmentations if and only if one has v ◦ τ = 0 for thecorresponding twisting cochain τ .Given a CDG-coalgebra C with a k -linear section w : k −→ C , set B = Cob w ( C )and τ = τ C,w : C −→ B . Theorem. (a)
The functors C ⊗ τ − : Hot ( B – mod ) −→ Hot ( C – comod ) and B ⊗ τ − : Hot ( C – comod ) −→ Hot ( B – mod ) induce functors D co ( B – mod ) −→ D co ( C – comod ) and D co ( C – comod ) −→ D co ( B – mod ) , which are mutually inverse equivalences oftriangulated categories. (b) The functors
Hom τ ( C, − ) : Hot ( B – mod ) −→ Hot ( C – contra ) and Hom τ ( B, − ) : Hot ( C – contra ) −→ Hot ( B – mod ) induce functors D ctr ( B – mod ) −→ D ctr ( C – contra ) and D ctr ( C – contra ) −→ D ctr ( B – mod ) , which are mutually inverse equivalences oftriangulated categories. (c) The above equivalences of triangulated categories D abs ( B – mod ) ≃ D co ( C – comod ) and D abs ( B – mod ) ≃ D ctr ( C – contra ) form a commutative diagram with the equivalenceof triangulated categories D co ( C – comod ) ≃ D ctr ( C – contra ) provided by the derivedfunctors L Φ C and R Ψ C of Theorem 5.2. Whenever C is an coaugmented CDG-coalgebra and accordingly B is a DG-algebra,the above equivalences of triangulated categories transform the trivial CDG-comodule k over C into the free DG-module B over B and the trivial CDG-contramodule k over C into the cofree DG-module Hom k ( B, k ) over B . Whenever C is a DG-coalgebra andaccordingly B is an augmented CDG-algebra, the same equivalences of triangulatedcategories transform the trivial CDG-module k over B into the cofree DG-comodule C over C and into the free DG-contramodule Hom k ( C, k ) over C . roof. The assertions about existence of induced functors in (a) and (b) hold for anyCDG-coalgebra C , CDG-algebra B , and twisting cochain τ : C −→ B . Indeed, thefunctors M C ⊗ τ M and N B ⊗ τ N send coacyclic objects to contractible ones,while the functors P Hom τ ( C, P ) and Q Hom τ ( B, Q ) send contraacyclicobjects to contractible ones, for the reasons explained in the proof of Theorem 6.3.One also finds that the induced functors are adjoint to each other, as explained inthe proof of Theorem 6.5. Now when B = Cob w ( C ) and τ = τ C,w , the adjunctionmorphisms are isomorphisms, as it was shown in the proof of Theorem 6.4. Noticethat D co ( B – mod ) = D abs ( B – mod ) = D ctr ( B – mod ) in this case by Theorem 3.6. Theproof of part (c) is identical to that of Theorem 6.3(c). (cid:3) We continue to assume that B = Cob w ( C ) and τ = τ C,w . Whenever C is a coaug-mented CDG-coalgebra, for any left CDG-comodule N over C the complex B ⊗ τ N computes Cotor C ( k, N ) ≃ Ext C ( k, N ) and for any left CDG-contramodule Q over C the complex Hom τ ( B, Q ) computes Coext C ( k, Q ) ≃ Ctrtor C ( k, Q ), as explainedin 6.5. The DG-algebra B itself, considered as a complex, computes Cotor C ( k, k ) ≃ Ext C ( k, k ). Whenever C is a DG-coalgebra, for any left CDG-module M over B thecomplex C ⊗ τ M computes Tor B,II ( k, M ) and for any left CDG-module P over B thecomplex Hom τ ( C, P ) computes Ext
IIB ( k, P ). The DG-coalgebra C itself, consideredas a complex, computes Tor B,II ( k, k ). Corollary.
Let C be a conilpotent CDG-coalgebra, w : k −→ C be its coaugmentation,and A = Cob w ( C ) be its cobar-construction. Then the derived category D ( A – mod ) and the absolute derived category D abs ( A – mod ) coincide; in other words, any acyclicDG-module over A is absolutely acyclic.Proof. Compare Theorem 6.4 and Theorem 6.7. (cid:3)
For more general results of this kind, see 6.8 and 9.4. For a counterexample showingthat the conilpotency condition is necessary in this Corollary, see Remark 6.10.Now let τ : C −→ B be any twisting cochain between a CDG-coalgebra C and aCDG-algebra B . Let us discuss the adjunction properties of our functors betweenthe homotopy categories in some more detail. Notice that the functor M C ⊗ τ M is the composition of left adjoint functors M Hom τ ( C, M ) and Q Φ C ( Q ).So the corresponding composition of right adjoint functors N Hom τ ( B, Ψ C ( N ))is right adjoint to the functor M C ⊗ τ M . At the same time, the functor N B ⊗ τ N is left adjoint to the functor M C ⊗ τ M . Analogously, the functor M Hom τ ( C, M ) is the composition of right adjoint functors M C ⊗ τ M and N Ψ C ( N ). So the corresponding composition of left adjoint functors Q B ⊗ τ Φ C ( Q ) is left adjoint to the functor M Hom τ ( C, M ). At the same time, thefunctor Q Hom τ ( B, Q ) is right adjoint to the functor M Hom τ ( C, M ). .8. Koszul cogenerators.
Let C be a CDG-coalgebra over a field k . Suppose C isendowed with a decreasing filtration by graded subspaces C = G C ⊃ G C ⊂ G C ⊃· · · ⊃ G N C ⊃ G N +1 C = 0 such that G is compatible with the comultiplication andthe counit, d C ( G n C ) ⊂ G n − C , and h C ( G C ) = 0. Let gr G C = L n G n C/G n +1 C bethe associated quotient coalgebra; it is a bigraded coalgebra with a grading i inducedby the grading of C and a grading 0 n N coming from the filtration G . Assumethat the coalgebra gr G C is Koszul [47, 48] in its nonnegative grading n .The differential d C on C induces a differential d gr C on gr G C having the degrees i = 1and n = −
1. The linear function h C on C induces a linear function h gr C on gr G C having the degrees i = 2 and n = −
2. This defines a structure of CDG-coalgebra(with respect to the grading i ) on gr G C . The natural map k ≃ C/G G −→ gr G C provides a coaugmentation w for gr G C . Consider the DG-algebra Cob w (gr G C ). Theincreasing filtration F on gr G C coming from the grading n is compatible with theCDG-coalgebra structure and induces an increasing filtration F on Cob w (gr G C ).Considering the associated graded DG-algebra gr F Cob w (gr G C ), one can show [46,Section 7 of Chapter 5] that the cohomology algebra B of the DG-algebra Cob w (gr G C )is concentrated in degree 0 with respect to the difference of the grading n inducedby the grading n of gr G C and the grading m of Cob w (gr G C ) by the tensor degrees.Moreover, the associated graded algebra gr F B is quadratic dual to the Koszul coalge-bra gr G C . The component F B is identified with the direct sum k ⊕ V of the subspace k = F B and a graded subspace V ⊂ B naturally isomorphic to ( G C/G C )[ − k -linear section w : k −→ C of the counit map of C . De-fine a graded algebra morphism Cob w ( C ) −→ B by the rule that the map C + [ − −→ B is the composition C + [ −
1] = G C [ − −→ ( G C/G C )[ − ≃ V −→ B . We claimthat the CDG-algebra structure on Cob w ( C ) induces such a structure on B . Indeed,the graded algebra B is the quotient algebra of the graded algebra Cob w ( C ) by theideal generated by G C [ − d Cob ( G C [ − d ( F n B ) ⊂ F n +1 B and h ∈ F B . The associated quotient algebra gr F B is Koszul with respect to thegrading n coming from the filtration F . In addition, gr F B has a finite homologicaldimension as a graded algebra with respect to the grading n , or equivalently, as abigraded algebra. Conversely, given a CDG-algebra B with an increasing filtration F with the above properties one can recover a CDG-coalgebra C . One proves thisusing the dual version of the spectral sequence argument from [46], which works inthe assumption of finite homological dimension.The graded algebra B has a finite (left and right) homological dimension, sinceone can compute the spaces Ext over it in terms of nonhomogeneous Koszul com-plexes. Let τ : C −→ B be the twisting cochain corresponding to the morphism ofCDG-algebras Cob w ( C ) −→ B . Theorem.
The assertions (a-c) of Theorem 6.7 hold for the CDG-coalgebra C , theDG-algebra B , and the twisting cochain τ . roof. Let us check that the adjunction morphism N −→ C ⊗ τ B ⊗ τ N has a coacycliccone for any CDG-comodule N over C . Introduce an increasing filtration F on thiscone K by the rules F − K = 0, F − K = N [1], and F n K = cone( N → C ⊗ k F n B ⊗ k N )for n >
0. The differential d on K does not preserve this filtration; still one has d ( F n K ) ⊂ F n +1 K . Let ∂ denote the induced differential on the associated quotient C -comodule gr F K . Then (gr F K, ∂ ) is an exact complex of graded C -comodules;indeed, it is isomorphic to the nonhomogeneous Koszul resolution of the graded C -comodule N . Consider the filtration of the CDG-comodule K over C by theCDG-subcomodules F n K + d ( F n K ). Then the associated quotient CDG-comodulesof this filtration are contractible, hence K is coacyclic. To check that the adjunctionmorphism B ⊗ τ C ⊗ τ M −→ M has an absolutely acyclic cone for any CDG-module M over B , one argues in the analogous way. Introduce a decreasing filtration G onthis cone K by the rules K/G − K = 0, K/G K = M , and K/G n +1 K = cone( B ⊗ τ C/G n +1 C ⊗ τ M → M ) for n >
0. Then the associated quotient CDG-modules of thefinite filtration of K by the CDG-submodules G n K + d ( G n K ) are contractible. Therest is explained in the proof of Theorem 6.7 and the previous theorems. (cid:3) Whenever C is a coaugmented coalgebra and w is its coaugmentation, theCDG-algebra B is, in fact, a DG-algebra. In this case, for any left CDG-comodule N over C the complex B ⊗ τ N computes Cotor C ( k, N ) ≃ Ext C ( k, N ) and for any leftCDG-contramodule Q over C the complex Hom τ ( B, Q ) computes Coext C ( k, Q ) ≃ Ctrtor C ( k, Q ), as explained in 6.5. The DG-algebra B itself, considered as a com-plex, computes Cotor C ( k, k ) ≃ Ext C ( k, k ). Whenever C is, actually, a DG-coalgebra,the CDG-algebra B is augmented. In this case, for any left CDG-module M over B the complex C ⊗ τ M computes Tor B,II ( k, M ) and for any left CDG-module P over B the complex Hom τ ( C, P ) computes Ext
IIB ( k, P ). The DG-coalgebra C itself,considered as a complex, computes Tor B,II ( k, k ).Let Hot ( B – mod fgp ) denote the full triangulated subcategory of the homotopy cat-egory Hot ( B – mod ) consisting of all CDG-modules that are projective and finitelygenerated as graded B -modules. The category Hot ( B – mod fgp ) can be also consideredas a full triangulated subcategory of D co ( B – mod ) = D ctr ( B – mod ) = D abs ( B – mod ). Corollary 1.
For a CDG-algebra B as above, the objects of the triangulated sub-category Hot ( B – mod fgp ) ⊂ D abs ( B – mod ) are compact generators of the triangulatedcategory D abs ( B – mod ) .Proof. It was shown in 5.5 (see also 4.6) that the objects of the full triangulated sub-category D abs ( C – comod fd ) ⊂ D co ( C – comod ) are compact generators of the coderivedcategory D co ( C – comod ). Obviously, the Koszul duality functor N B ⊗ τ M fromthe above Theorem maps D abs ( C – comod fd ) into Hot ( B – mod fgp ) ⊂ D abs ( B – mod ). Onthe other hand, by Theorem 3.6 the absolute derived category D abs ( B – mod ) is equiv-alent to the homotopy category Hot ( B – mod proj ) of CDG-modules that are projective s graded B -modules. It is clear that the objects of Hot ( B – mod fgp ) are compact in Hot ( B – mod proj ). The assertion of Corollary follows from that of Theorem, part (a)together with these observations. (cid:3) For a discussion of questions and results related to Corollary 1, see Question 3.8.
Corollary 2.
In the above notation, suppose that C is a conilpotent CDG-coalgebraand w : k −→ C is its coaugmentation. Then the derived category D ( B – mod ) andthe absolute derived category D abs ( B – mod ) coincide; in other words, any acyclicDG-module over B is absolutely acyclic.Proof. It suffices to show that the CDG-comodule C ⊗ τ M over C is coacyclic whenevera DG-module M over B is acyclic. It was explained in the proof of Theorem 6.4 howto do so. (cid:3) For other results similar to that of Corollary 2, see 9.4.6.9.
Cotor and Tor, Coext and Ext, Ctrtor and Tor; restriction and ex-tension of scalars.
Let C be a conilpotent CDG-coalgebra, A be a DG-algebra,and τ : C −→ A be an acyclic twisting cochain. Notice that by the right version ofTheorem 6.5 the functors M M ⊗ τ C and N N ⊗ τ A induce an equivalenceof triangulated categories D ( mod – A ) ≃ D co ( comod – C ). Theorem 1. (a)
The equivalences of triangulated categories D co ( comod – C ) ≃ D ( mod – A ) and D co ( C – comod ) ≃ D ( A – mod ) transform the functor Cotor C into thefunctor Tor A . (b) The equivalences of triangulated categories D co ( C – comod ) ≃ D ( A – mod ) and D ctr ( C – contra ) ≃ D ( A – mod ) transform the functor Coext C into the functor Ext A . (c) The equivalences of triangulated categories D co ( comod – C ) ≃ D ( mod – A ) and D ctr ( C – contra ) ≃ D ( A – mod ) transform the functor Ctrtor C into the functor Tor A .Proof. To prove part (a), it suffices to use either of the natural isomorphisms ofcomplexes N ′ (cid:3) C ( C ⊗ τ M ′′ ) ≃ ( N ′ ⊗ τ A ) ⊗ A M ′′ or ( M ′ ⊗ τ C ) (cid:3) C N ′′ ≃ M ′ ⊗ A ( A ⊗ τ N ′′ ).To check that one obtains the same isomorphism of functors in these two ways,notice that the two compositions N ′ (cid:3) C N ′′ −→ N ′ (cid:3) C ( C ⊗ τ A ⊗ τ N ′′ ) ≃ ( N ′ ⊗ τ A ) ⊗ A ( A ⊗ τ N ′′ ) and N ′ (cid:3) C N ′′ −→ ( N ′ ⊗ τ A ⊗ τ C ) (cid:3) C N ′′ −→ ( N ′ ⊗ τ A ) ⊗ A ( A ⊗ τ N ′′ ) coincide. To prove (b), use either of the isomorphisms Cohom C ( C ⊗ τ M, Q ) ≃ Hom A ( M, Hom τ ( A, Q )) or Cohom C ( N, Hom τ ( C, P )) ≃ Hom A ( A ⊗ τ N, P ).Alternatively, use the result of 5.3 and Theorem 6.5. To check (c), use the naturalisomorphism ( N ⊗ τ A ) ⊗ A P ≃ N ⊙ C Hom τ ( C, P ). (cid:3) Let C and D be conilpotent CDG-coalgebras, A and B be DG-algebras, τ : C −→ A and σ : D −→ B be acyclic twisting cochains. Let f : A −→ B be a morphism ofDG-algebras and g = ( g, a ) : C −→ D be a morphism of CDG-coalgebras. Assume hat the following commutativity equation holds: the difference σg − f τ is equal tothe composition of the map a : C −→ k and the unit map k −→ B . Proposition. (a)
The equivalences of triangulated categories D ( A – mod ) ≃ D co ( C – comod ) and D ( B – mod ) ≃ D co ( D – comod ) transform the functor I R f into the functor R E g and the functor L E f into the functor I R g . (b) The equivalences of triangulated categories D ( A – mod ) ≃ D ctr ( C – contra ) and D ( B – mod ) ≃ D ctr ( D – contra ) transform the functor I R f into the functor L E g andthe functor R E f into the functor I E g .Proof. To prove part (a), use the natural isomorphisms E g ( D ⊗ σ M ) ≃ C ⊗ τ R f ( M )and E f ( A ⊗ τ N ) ≃ B ⊗ σ R g ( N ) for a DG-module M over B and a CDG-comodule N over C . The proof of part (b) is similar. (cid:3) Now let C be a CDG-algebra endowed with a k -linear section w : k −→ C and B = Cob w ( C ) be its cobar-construction. More generally, one can assume that C and B are a CDG-coalgebra and a CDG-algebra related by the construction of 6.8. By theright version of Theorem 6.7 or 6.8, the functors M M ⊗ τ C and N N ⊗ τ B induce an equivalence of triangulated categories D abs ( mod – B ) ≃ D co ( comod – C ). Theorem 2. (a)
The equivalences of triangulated categories D co ( comod – C ) ≃ D abs ( mod – B ) and D co ( C – comod ) ≃ D abs ( B – mod ) transform the functor Cotor C intothe functor Tor
B,II . (b) The equivalences of triangulated categories D co ( C – comod ) ≃ D abs ( B – mod ) and D ctr ( C – contra ) ≃ D abs ( B – mod ) transform the functor Coext C into the functor Ext
IIB . (c) The equivalences of triangulated categories D co ( comod – C ) ≃ D ( mod – B ) and D ctr ( C – contra ) ≃ D ( B – mod ) transform the functor Ctrtor C into the functor Tor
B,II .Proof.
See the proof of Theorem 1. (cid:3)
Bar duality between algebras and coalgebras.
Graded tensor coalgebrasare cofree objects in the category of conilpotent graded coalgebras. More precisely,for any conilpotent graded coalgebra C with the coaugmentation w : k −→ C and anygraded vector space U there is a bijective correspondence between graded coalgebramorphisms C −→ L ∞ n =0 U ⊗ n and homogeneous k -linear maps C/w ( k ) −→ U ofdegree zero. Notice that the graded tensor coalgebra L ∞ n =0 U ⊗ n is conilpotent andany morphism of conilpotent graded coalgebras preserves the coaugmentations. Letus emphasize that the above assertion in not true when the graded coalgebra C isnot conilpotent.Let B be a CDG-algebra and v : B −→ k be a homogeneous k -linear retraction.Let C be a CDG-coalgebra that is conilpotent as a graded coalgebra; denote by w : k −→ C the coaugmentation map (which does not have to be a coaugmentationof C as a CDG-algebra). Then there is a natural bijective correspondence between orphisms of CDG-coalgebras C −→ Bar v ( B ) and twisting cochains τ : C −→ B such that τ ◦ w = 0. Whenever C is a DG-coalgebra and v is an augmentation of B ,a CDG-coalgebra morphism C −→ Bar v ( B ) is actually a morphism of DG-coalgebrasif and only if one has v ◦ τ = 0 for the corresponding twisting cochain τ .Let k – coalg conilpcdg denote the category of conilpotent CDG-coalgebras, k – coalg conilpdg denote the category of conilpotent DG-coalgebras, k – alg + dg denote the category ofDG-algebras with nonzero units, and k – alg augdg denote the category of augmentedDG-algebras (over the ground field k ). It follows from the above that the functor ofconilpotent cobar-construction Cob w : k – coalg conilpcdg −→ k – alg + dg is left adjoint to thefunctor of DG-algebra bar-construction Bar v : k – alg + dg −→ k – coalg conilpcdg . Analogously,the functor of conilpotent DG-coalgebra cobar-construction Cob w : k – coalg conilpdg −→ k – alg augdg is right adjoint to the functor of augmented DG-algebra bar-constructionBar v : k – alg augdg −→ k – coalg conilpdg .A morphism of conilpotent CDG-coalgebras ( f, a ) : C −→ D is called a filteredquasi-isomorphism if there exist increasing filtrations F on C and D satisfying thefollowing conditions. The filtrations F must be compatible with the comultiplicationsand differentials on C and D ; one must have F C = w C ( k ) and F D = w D ( k ), so that,in particular, the associated quotient objects gr F C and gr F D are DG-coalgebras;and the induced morphism gr F f : gr F C −→ gr F D must be a quasi-isomorphismof graded complexes of vector spaces. A morphism of conilpotent DG-coalgebrasis a filtered quasi-isomorphism if it is a filtered quasi-isomorphism as a morphismof conilpotent CDG-coalgebras. The classes of filtered quasi-isomorphisms will bedenoted by FQuis ⊂ k – coalg conilpcdg and FQuis ⊂ k – coalg conilpdg . Let us emphasize thatthere is no claim that the classes of filtered quasi-isomorphisms are closed undercomposition of morphisms. The classes of quasi-isomorphisms of DG-algebras andaugmented DG-algebras will be denoted by Quis ⊂ k – alg + dg and Quis ⊂ k – alg augdg . Theorem. (a)
The functors
Cob w : k – coalg conilpcdg −→ k – alg + dg and Bar v : k – alg + dg −→ k – coalg conilpcdg induce functors between the localized categories k – coalg conilpcdg [ FQuis − ] −→ k – alg + dg [ Quis − ] and k – alg + dg [ Quis − ] −→ k – coalg conilpcdg [ FQuis − ] , which are mutuallyinverse equivalences of categories. (b) The functors
Cob w : k – coalg conilpdg −→ k – alg augdg and Bar v : k – alg augdg −→ k – coalg conilpdg induce functors between the localized categories k – coalg conilpdg [ FQuis − ] −→ k – alg augdg [ Quis − ] and k – alg augdg [ Quis − ] −→ k – coalg conilpdg [ FQuis − ] , which are mutuallyinverse equivalences of categories.Proof. We will prove part (a); the proof of part (b) is similar. For any filteredquasi-isomorphism of conilpotent CDG-coalgebras ( f, a ) : C −→ D the induced orphism of cobar-constructions Cob w ( f, a ) : Cob w ( C ) −→ Cob w ( D ) is a quasi-isomorphism of DG-algebras. Indeed, let F denote the increasing filtrations onthe cobar-constructions induced by the filtrations F of C and D ; then the mor-phism of associated graded DG-algebras gr F Cob w ( f, a ) is a quasi-isomorphism,since the tensor products and the cones of morphisms of complexes preserve quasi-isomorphisms. Conversely, for any quasi-isomorphism of DG-algebras g : A −→ B the induced morphism of bar-constructions Bar v ( g ) : Bar v ( A ) −→ Bar v ( B ) is a fil-tered quasi-isomorphism. Indeed, it suffices to consider the increasing filtrationsof bar-constructions associated with their nonnegative gradings n by the numberof factors in tensor powers. So the induced functors exist; it remains to checkthat they are mutually inverse equivalences. For any DG-algebra A , the adjunc-tion morphism Cob w (Bar v ( A )) −→ A is a quasi-isomorphism. One can prove thisby passing to the associated quotients with respect to the increasing filtration F on A defined by the rules F A = k and F A = A , and the induced filtrationon Cob w (Bar v ( A )). Finally, for any conilpotent CDG-coalgebra C , the adjunctionmorphism C −→ Bar v (Cob w ( C )) is a filtered quasi-isomorphism. Indeed, considerthe natural increasing filtration F on C defined in 6.4 and the induced filtration F on Bar v (Cob w ( C )). We have to prove that our adjunction morphism becomesa quasi-isomorphism after passing to the associated quotient objects, i. e., the mor-phism of graded DG-coalgebras gr F C −→ Bar v (Cob w (gr F C )) is a quasi-isomorphism.Here it suffices to consider the decreasing filtration G on gr F C defined by the rules G gr F C = gr F C , G gr F C = ker( ε : gr F C → k ), and G gr F C = 0. The inducedfiltration on Bar v (Cob w (gr F C )) stabilizes at every degree of the nonnegative gradingcoming from the filtration F and the morphism gr F C −→ Bar v (Cob w (gr F C )) can beeasily seen to become a quasi-isomorphism after passing to the associated quotientobjects with respect to the filtration G . (cid:3) Remark.
Notice that the notion of a filtered quasi-isomorphism makes sense forconilpotent CDG-coalgebras only, as any CDG-coalgebra admitting an increasing fil-tration F satisfying the conditions in the definition of a filtered quasi-isomorphismis conilpotent. And one cannot even speak about conventional (nonfiltered) quasi-isomorphisms of CDG-coalgebras, as the latter are not complexes. Furthermore,the assertions of Theorem do not hold with the filtered quasi-isomorphisms replacedwith conventional quasi-isomorphisms of DG-coalgebras, or with the conilpotencycondition dropped. Indeed, let A be any DG-algebra; consider it as a DG-algebrawithout unit and add a unit formally to it, obtaining an augmented DG-algebra k ⊕ A with the augmentation v . Then the morphisms of augmented DG-algebras k −→ k ⊕ A −→ k induce quasi-isomorphisms of bar-constructions Bar v ( k ) −→ Bar v ( k ⊕ A ) −→ Bar v ( k ); applying the cobar-construction, we find that the mor-phisms Cob w (Bar v ( k )) −→ Cob w (Bar v ( k ⊕ A )) −→ Cob w (Bar v ( k )) are not quasi-isomorphisms, since the middle term is quasi-isomorphic to k ⊕ A . Analogously, let D e any DG-coalgebra; consider it as a DG-coalgebra without counit and add a counitformally to it, obtaining a coaugmented DG-coalgebra k ⊕ D with the coaugmenta-tion w . Then the morphisms of augmented DG-algebras k −→ Cob w ( k ⊕ D ) −→ k arequasi-isomorphisms and it follows that the induced morphisms of bar-constructionsBar v ( k ) −→ Bar v (Cob w ( k ⊕ D )) −→ Bar v ( k ) are also quasi-isomorphisms, hence thecohomology of the DG-coalgebra Bar v (Cob w ( k ⊕ D )) is different from that of k ⊕ D .And there is even no natural morphism between C and Bar v (Cob w ( C )) for a non-conilpotent DG-coalgebra C . Finally, let D be a DG-coalgebra and N be a leftDG-comodule over D . Consider N as a DG-comodule over the above DG-coalgebra k ⊕ D ; then the DG-module Cob w ( k ⊕ D ) ⊗ τ k ⊕ D,w N over Cob w ( k ⊕ D ) is acyclic. Itfollows that the assertions of Theorem 6.4 and Corollary 6.7 do not hold without theconilpotency assumption on the coaugmented CDG-coalgebra C .7. A ∞ -Algebras and Curved A ∞ -Coalgebras Nonunital A ∞ -algebras. Let A be a graded vector space over a field k . Con-sider the graded tensor coalgebra (cofree conilpotent coassociative graded coalgebra) L ∞ n =0 A [1] ⊗ n with its coaugmentation w : k ≃ A [1] ⊗ −→ L n A [1] ⊗ n . A nonuni-tal A ∞ -algebra structure on A is, by the definition, a coaugmented DG-coalgebrastructure on L n A [1] ⊗ n , i. e., an odd coderivation d of degree 1 on L n A [1] ⊗ n suchthat d = 0 and d ◦ w = 0. Since a coderivation of L n A [1] ⊗ n is uniquely de-termined by its composition with the projection L n A [1] ⊗ n −→ A [1] ⊗ ≃ A [1], anonunital A ∞ -algebra structure on A can be considered as a sequence of linear maps m n : A ⊗ n −→ A , n = 1, 2, . . . of degree 2 − n . More precisely, define the maps m n by the rule that the image of the element d ( a ⊗ · · · ⊗ a n ) under the projection to A equals ( − n + P nj =1 ( n − j )( | a j | +1) m n ( a ⊗ · · · ⊗ a n ) for a j ∈ A . The sequence of maps m n must satisfy a sequence of quadratic equations corresponding to the equation d = 0on the coderivation d . We will not write down these equations explicitly.A morphism of nonunital A ∞ -algebras f : A −→ B over k is, by the definition,a morphism of (coaugmented) DG-coalgebras L n A [1] ⊗ n −→ L n B [1] ⊗ n . Since agraded coalgebra morphism into a graded tensor coalgebra L n B [1] ⊗ n is determinedby its composition with the projection L n B [1] ⊗ n −→ B [1] ⊗ ≃ B [1] and any mor-phism of conilpotent graded coalgebras preserves coaugmentations, a morphism ofnonunital A ∞ -algebras f : A −→ B can be considered as a sequence of linear maps f n : A ⊗ n −→ B , n = 1, 2, . . . of degree 1 − n . More precisely, define the maps f n by the rule that the image of the element f ( a ⊗ · · · ⊗ a n ) under the projection to B equals ( − n − P nj =1 ( n − j )( | a j | +1) f n ( a ⊗ · · · ⊗ a n ) for a j ∈ A . The sequence of maps f n must satisfy a sequence of polynomial equations corresponding to the equation d ◦ f = f ◦ d on the morphism f . et A be a nonunital A ∞ -algebra over a field k and M be a graded vector spaceover k . A structure of nonunital left A ∞ -module over A on M is, by the definition,a structure of DG-comodule over the DG-coalgebra L n A [1] ⊗ n on the cofree gradedleft comodule L n A [1] ⊗ n ⊗ k M over the graded coalgebra L n A [1] ⊗ n . Analogously, astructure of nonunital right A ∞ -module over A on a graded vector space N is definedas a structure of DG-comodule over L n A [1] ⊗ n on the cofree graded right comodule N ⊗ k L n A [1] ⊗ n . Since a coderivation of a cofree graded comodule L n A [1] ⊗ n ⊗ k M compatible with a given coderivation of the graded coalgebra L n A [1] ⊗ n is determinedby its composition with the projection L n A [1] ⊗ n ⊗ k M −→ M induced by thecounit map L n A [1] ⊗ n −→ k , a nonunital left A ∞ -module structure on M can beconsidered as a sequence of linear maps l n : A ⊗ n ⊗ M −→ M , n = 0, 1, . . . of degree1 − n . More precisely, define the maps l n by the rule that the image of the element d ( a ⊗ · · · ⊗ a n ⊗ x ) under the projection to M equals ( − n + P nj =1 ( n − j )( | a j | +1) l n ( a ⊗· · · ⊗ a n ⊗ x ) for a j ∈ A and x ∈ M . The sequence of maps l n must satisfy a systemof nonhomogeneous quadratic equations corresponding to the equation d = 0 onthe coderivation d on L n A [1] ⊗ n ⊗ k M . Analogously, a nonunital right A ∞ -modulestructure on N can be considered as a sequence of linear maps r n : N ⊗ A ⊗ n −→ N defined by the rule that the image of the element d ( y ⊗ a ⊗ · · · ⊗ a n ) under theprojection N ⊗ k L n A [1] ⊗ n −→ N equals ( − n | y | + P nj =1 ( n − j )( | a j | +1) r n ( y ⊗ a ⊗ · · · ⊗ a n )for a j ∈ A and y ∈ N .The complex of morphisms between nonunital left A ∞ -modules L and M over anonunital A ∞ -algebra A is, by the definition, the complex of morphisms betweenleft DG-comodules L n A [1] ⊗ n ⊗ k L and L n A [1] ⊗ n ⊗ k M over the DG-coalgebra L n A [1] ⊗ n . Analogously, the complex of morphisms between nonunital rightA ∞ -modules R and N over A is, by the definition, the complex of morphisms betweenright DG-comodules R ⊗ k L n A [1] ⊗ n and N ⊗ k L n A [1] ⊗ n over the DG-coalgebra L n A [1] ⊗ n . A morphism of nonunital left A ∞ -modules f : L −→ M of degree i isthe same that a sequence of linear maps f n : A ⊗ n ⊗ k L −→ M , n = 0, 1, . . . ofdegree i − n . More precisely, define the maps f n by the rule that the image of theelement f ( a ⊗ · · · ⊗ a n ⊗ x ) under the projection L n A [1] ⊗ n ⊗ k M −→ M equals( − n + P nj =1 ( n − j )( | a j | +1) f n ( a ⊗ · · · ⊗ a n ⊗ x ) for a j ∈ A and x ∈ L . Any sequenceof linear maps f n corresponds to a (not necessarily closed) morphism of nonunitalA ∞ -modules f . Analogously, a morphism of nonunital right A ∞ -modules g : R −→ N of degree i is the same that a sequence of linear maps g n : A ⊗ n ⊗ k R −→ N of de-gree i − n . More precisely, define the maps g n by the rule that the image of theelement g ( y ⊗ a ⊗ · · · ⊗ a n ) under the projection N ⊗ k L n A [1] ⊗ n −→ N equals( − n | y | + P nj =1 ( n − j )( | a j | +1) g n ( y ⊗ a ⊗ · · · ⊗ a n ) for a j ∈ A and y ∈ R .For any CDG-coalgebra C , the functors Φ C and Ψ C of 5.1–5.2 provide an equiv-alence between the DG-category of left CDG-comodules over C that are cofree asgraded C -comodules and the DG-category of left CDG-contramodules over C that re free as graded C -contramodules. So one can alternatively define a nonunital leftA ∞ -module M over a nonunital A ∞ -algebra A as a graded vector space for which astructure of DG-contramodule over the DG-coalgebra L n A [1] ⊗ n is given on the freegraded contramodule Hom k ( L n A [1] ⊗ n , M ) over the graded coalgebra L n A [1] ⊗ n .Since a contraderivation of a free graded contramodule Hom k ( L n A [1] ⊗ n , M ) com-patible with a given coderivation of the graded coalgebra L n A [1] ⊗ n is determinedby its restriction to the graded subspace M ⊂ Hom k ( L n A [1] ⊗ n , M ), a nonunital leftA ∞ -module structure on M can be considered as a sequence of linear maps p n : M −→ Hom k ( A ⊗ n , M ), n = 0, 1, . . . of degree 1 − n . More precisely, define the maps p n bythe formula p n ( x )( a ⊗· · ·⊗ a n ) = ( − n + n | x | + P nj =1 ( n − j )( | a j | +1) d ( x )( a ⊗· · ·⊗ a n ) for a j ∈ A and x ∈ M . Then the maps p n are related to the above maps l n : A ⊗ n ⊗ k M −→ M by the rule p n ( x )( a ⊗ · · · ⊗ a n ) = ( − | x | P nj =1 | a j | l n ( a ⊗ · · · ⊗ a n ⊗ x ).Furthermore, one can alternatively define the complex of morphisms betweennonunital left A ∞ -modules L and M over a nonunital A ∞ -algebra A as thecomplex of morphisms between left DG-contramodules Hom k ( L n A [1] ⊗ n , L ) andHom k ( L n A [1] ⊗ n , M ) over the DG-coalgebra L n A [1] ⊗ n . Thus a (not necessar-ily closed) morphism of nonunital left A ∞ -modules f : L −→ M of degree i isthe same that a sequence of linear maps f n : L −→ Hom k ( A ⊗ n , M ), n = 0,1, . . . of degree i − n . More precisely, define the maps f n by the formula f n ( x )( a ⊗ · · · ⊗ a n ) = ( − n + n | x | + P nj =1 ( n − j )( | a j | +1) f ( x )( a ⊗ · · · ⊗ a n ) for a j ∈ A and x ∈ L , where L is considered as a graded subspace in Hom k ( L n A [1] ⊗ n , L ).Then the maps f n are related to the above maps f n : A ⊗ n ⊗ k L −→ M by the rule f n ( x )( a ⊗ · · · ⊗ a n ) = ( − | x | P nj =1 | a j | f n ( a ⊗ · · · ⊗ a n ⊗ x ).7.2. Strictly unital A ∞ -algebras. Let A be a nonunital A ∞ -algebra over a field k .An element 1 ∈ A of degree 0 is called a strict unit if one has m (1 ⊗ a ) = a = m ( a ⊗ a ∈ A and m n ( a ⊗ · · · ⊗ a j − ⊗ ⊗ a j +1 ⊗ · · · ⊗ a n ) = 0 for all n = 2,1 j n , and a t ∈ A . Obviously, a strict unit is unique if it exists. A strictly unital A ∞ -algebra is a nonunital A ∞ -algebra that has a strict unit. A morphism of strictlyunital A ∞ -algebras f : A −→ B is a morphism of nonunital A ∞ -algebras such that f (1 A ) = 1 B and f n ( a ⊗ · · · ⊗ a j − ⊗ A ⊗ a j +1 ⊗ · · · ⊗ a n ) = 0 for all n > a t ∈ A .Notice that for a strictly unital A ∞ -algebra A with the unit 1 A one has 1 A = 0 if andonly if A = 0. We will assume our strictly unital A ∞ -algebras to have nonzero units.A strictly unital left A ∞ -module M over a strictly unital A ∞ -algebra A is a nonuni-tal left A ∞ -module such that l (1 ⊗ x ) = x and l n ( a ⊗ · · · ⊗ a j − ⊗ ⊗ a j +1 ⊗ · · · ⊗ a n ⊗ x ) = 0 for all n >
1, 1 j n , a t ∈ A , and x ∈ M . Equivalently, one musthave p ( x )(1) = x and p n ( x )( a ⊗ · · · ⊗ a j − ⊗ ⊗ a j +1 ⊗ · · · ⊗ a n ) = 0. Analogously, a strictly unital right A ∞ -module N over A is a nonunital right A ∞ -module such that r ( y ⊗
1) = y and r n ( y ⊗ a ⊗ · · · ⊗ a j − ⊗ ⊗ a j +1 ⊗ · · · ⊗ a j ) = 0 for all n > a t ∈ A , and y ∈ N . he complex of morphisms between strictly unital left A ∞ -modules L and M over astrictly unital A ∞ -algebra A is the subcomplex of the complex of morphisms between L and M as nonunital A ∞ -modules consisting of all morphisms f : L −→ M such that f n ( a ⊗ · · · ⊗ a j − ⊗ ⊗ a j +1 ⊗ · · · ⊗ a n ⊗ x ) = 0 for all n >
0, 1 j n , a t ∈ A , and x ∈ L . Equivalently, one must have f n ( x )( a ⊗ · · · ⊗ a j − ⊗ ⊗ a j +1 ⊗ · · · ⊗ a n ) = 0.Analogously, the complex of morphisms between strictly unital right A ∞ -modules R and N over A is the subcomplex of the complex of morphisms between R and N as nonunital A ∞ -modules consisting of all morphisms g : R −→ N such that g n ( y ⊗ a ⊗ · · · ⊗ a j − ⊗ ⊗ a j +1 ⊗ · · · ⊗ a n ) = 0 for all n > a t ∈ A , and y ∈ R .Let A be a nonunital A ∞ -algebra and 1 A ∈ A be a nonzero element of degree 0. Set A + = A/k · A . Then the graded tensor coalgebra L n A + [1] ⊗ n is a quotient coalgebraof the tensor coalgebra L n A [1] ⊗ n . Denote by K A the kernel of the natural surjection L n A [1] ⊗ n −→ L n A + [1] ⊗ n and by κ A : K A −→ k the homogeneous linear functionof degree 1 sending 1 A ∈ K A ∩ A [1] to 1 ∈ k and annihilating K A ∩ A [1] ⊗ n for all n >
1. Let θ A : L n A [1] ⊗ n −→ k be any homogeneous linear function of degree 1extending the linear function κ A on K A . Then the element 1 A ∈ A is a strict unit ifand only if the odd coderivation d ′ ( c ) = d ( c ) + θ A ∗ c − ( − | c | c ∗ θ A of degree 1 onthe tensor coalgebra L n A [1] ⊗ n preserves the subspace K A and the linear function h ′ ( c ) = θ A ( d ( c )) + θ A ( c ) of degree 2 on L n A [1] ⊗ n annihilates K A . This conditiondoes not depend on the choice of θ A . For strictly unital A ∞ -algebras A and B , amorphism of nonunital A ∞ -algebras f : A −→ B is a morphism of strictly unitalA ∞ -algebras if and only if f ( K A ) ⊂ K B and κ B ◦ f | K A = κ A .Let A be a strictly unital A ∞ -algebra and M be a nonunital left A ∞ -moduleover A . Then M is a strictly unital A ∞ -module if and only if the odd coderiva-tion d ′ ( z ) = d ( z ) + θ A ∗ z of degree 1 on the cofree comodule L n A [1] ⊗ n ⊗ k M compatible with the coderivation d ′ of the coalgebra L n A [1] ⊗ n preserves the sub-space K A ⊗ k M ⊂ L n A [1] ⊗ n ⊗ k M . Equivalently, the odd contraderivation d ′ ( q ) = d ( q ) + θ A ∗ q of degree 1 on the free contramodule Hom k ( L n A [1] ⊗ n , M ) compatiblewith the coderivation d ′ of the coalgebra L n A [1] ⊗ n must preserve the subspaceHom k ( L n A + [1] ⊗ n , M ) ⊂ Hom k ( L n A [1] ⊗ n , M ). Analogously, a nonunital rightA ∞ -module N over A is a strictly unital A ∞ -module if and only if the odd coderivation d ′ ( z ) = d ( z ) − ( − | z | z ∗ θ A of degree 1 on the cofree comodule N ⊗ k L n A [1] ⊗ n com-patible with the coderivation d ′ of the coalgebra L n A [1] ⊗ n preserves the subspace N ⊗ k K A ⊂ N ⊗ k L n A [1] ⊗ n . For strictly unital left A ∞ -modules L and M over A ,a (not necessarily closed) morphism of nonunital A ∞ -modules f : L −→ M is a mor-phism of strictly unital A ∞ -modules if and only if one has f ( K A ⊗ k L ) ⊂ K A ⊗ k M ,or equivalently, f (Hom k ( L n A + [1] ⊗ n , L )) ⊂ Hom k ( L n A + [1] ⊗ n , M ). Analogously,for strictly unital right A ∞ -modules R and N over A , a (not necessarily closed)morphism of nonunital A ∞ -modules g : R −→ N is a morphism of strictly unitalA ∞ -modules if and only if one has g ( R ⊗ k K A ) ⊂ N ⊗ k K A . et A be a strictly unital A ∞ -algebra. Identify k with the subspace k · A ⊂ A and choose a homogeneous k -linear retraction v : A −→ k . Define the homogeneouslinear function θ A : L n A [1] ⊗ n −→ k of degree 1 by the rules θ A ( a ) = v ( a ) and θ A ( a ⊗ · · · ⊗ a n ) = 0 for n = 1. Then the linear function θ A is an extension ofthe linear function κ A : K A −→ k . Let d : L n A + [1] ⊗ n −→ L n A + [1] ⊗ n be the mapinduced by the odd coderivation d ′ of L n A [1] ⊗ n defined by the above formula, andlet h : L n A + [1] ⊗ n −→ k be the linear function induced by the above linear func-tion h ′ . Then Bar v ( A ) = ( L n A + [1] ⊗ n , d, h ) is a coaugmented (and consequenly,conilpotent) CDG-coalgebra with the coaugmentation k ≃ A + [1] ⊗ −→ L n A + [1] ⊗ n .The CDG-coalgebra Bar v ( A ) is called the bar-construction of a strictly unitalA ∞ -algebra A .Let f : A −→ B be a morphism of strictly unital A ∞ -algebras. Let v : A −→ k and v : B −→ k be homogeneous k -linear retractions, and let θ A and θ B be the correspond-ing homogeneous linear functions of degree 1 on the graded tensor coalgebras. Themorphism of tensor coalgebras f : L n A [1] ⊗ n −→ L n B [1] ⊗ n maps K A into K B , so itinduces a morphism of graded tensor coalgebras L n A + [1] ⊗ n −→ L n B + [1] ⊗ n , whichwe will denote also by f . The linear function θ B ◦ f − θ A : L n A [1] ⊗ n −→ k annihilates K A , so it induces a linear function L n A + [1] ⊗ n −→ k , which we will denote by η f .Then the pair ( f, η f ) is a morphism of CDG-coalgebras Bar v ( A ) −→ Bar v ( B ). Thusthe bar-construction A Bar v ( A ) is a functor from the category of strictly uni-tal A ∞ -algebras with nonzero units to the category of coaugmented CDG-coalgebraswhose underlying graded coalgebras are graded tensor coalgebras. One can easily seethat this functor is an equivalence of categories. Alternatively, one can use any linearfunction θ A of degree 1 extending the linear function κ A in the construction of thisequivalence of categories.To obtain the inverse functor, assign to a conilpotent CDG-coalgebra ( D, d D , h D )the conilpotent DG-coalgebra ( C, d C ) constructed as follows. First, adjoin to D asingle cofree cogenerator of degree −
1, obtaining a conilpotent graded coalgebra C endowed with a graded coalgebra morphism C −→ D and a homogeneous linearfunction θ : C −→ k of degree 1. Second, define the odd coderivation d ′ C of degree 1on the graded coalgebra C by the conditions that d ′ C must preserve the kernel of thegraded coalgebra morphism C −→ D and induce the differential d D on D , and thatthe equation θ ( d ′ C ( c )) = θ ( c ) + h D ( c ) must hold for all c ∈ C , where h D is consideredas a linear function on C . Finally, set d C ( c ) = d ′ C ( c ) − θ ∗ c + ( − | c | c ∗ θ for all c ∈ C .Let A be a strictly unital A ∞ -algebra, v : A −→ k be a homogeneous k -linearretraction, and θ A : L n A [1] ⊗ n −→ k be the corresponding homogeneous linearfunction of degree 1. Let M be a strictly unital left A ∞ -module over A . Set d : L n A + [1] ⊗ n ⊗ k M −→ L n A + [1] ⊗ n ⊗ k M to be the map induced by the dif-ferential d ′ on L n A [1] ⊗ n ⊗ k M defined by the above formula. Then Bar v ( A, M ) =( L n A + [1] ⊗ n ⊗ k M, d ) is a left CDG-comodule over the CDG-coalgebra Bar v ( A ). urthermore, set d : Hom k ( L n A + [1] ⊗ n , M ) −→ Hom k ( L n A + [1] ⊗ n , M ) to be therestriction of the differential d ′ on Hom k ( L n A [1] ⊗ n , M ) defined above. ThenCob v ( A, M ) = (Hom k ( L n A + [1] ⊗ n , M ) , d ) is a left CDG-contramodule over theCDG-coalgebra Bar v ( A ). Analogously, for a strictly unital right A ∞ -module N over A set d : N ⊗ k L n A + [1] ⊗ n −→ N ⊗ k L n A + [1] ⊗ n to be the map induced by the differen-tial d ′ on N ⊗ k L n A [1] ⊗ n defined above. Then Bar v ( N, A ) = ( N ⊗ k L n A + [1] ⊗ n , d )is a right CDG-comodule over the CDG-coalgebra Bar v ( A ).To a (not necessarily closed) morphism of strictly unital left A ∞ -modules f : L −→ M over A one can assign the induced maps L n A + [1] ⊗ n ⊗ k L −→ L n A + [1] ⊗ n ⊗ k M and Hom k ( L n A + [1] ⊗ n , L ) −→ Hom k ( L n A + [1] ⊗ n , M ). These are a (not necessarilyclosed) morphism of CDG-comodules Bar v ( A, L ) −→ Bar v ( A, M ) and a (not neces-sarily closed) morphism of CDG-contramodules Cob v ( A, L ) −→ Cob v ( A, M ) over theCDG-coalgebra Bar v ( A ). So we obtain the DG-functor M Bar v ( A, M ), whichis an equivalence between the DG-category of strictly unital left A ∞ -modules over A and the DG-category of left CDG-comodules over Bar v ( A ) that are cofree as gradedcomodules, and the DG-functor M Cob v ( A, M ), which is an equivalence betweenthe DG-category of strictly unital left A ∞ -modules over A and the DG-category ofleft CDG-contramodules over Bar v ( A ) that are free as graded contramodules. Thesetwo equivalences of DG-categories form a commutative diagram with the equivalencebetween the DG-category of CDG-comodules that are cofree as graded comodulesand the DG-category of CDG-contramodules that are free as graded contramodulesprovided by the functors Ψ Bar v ( A ) and Φ Bar v ( A ) . Analogously, to a (not necessarilyclosed) morphism of strictly unital right A ∞ -modules g : R −→ N over A one canassign the induced map R ⊗ k L n A + [1] ⊗ n −→ N ⊗ k L n A + [1] ⊗ n . This is a (notnecessarily closed) morphism of CDG-comodules Bar v ( R, A ) −→ Bar v ( N, A ). TheDG-functor N Bar v ( N, A ) is an equivalence between the DG-category of strictlyunital right A ∞ -modules over A and the DG-category of right CDG-comodules overBar v ( A ) that are cofree as graded comodules.Now let A be a DG-algebra with nonzero unit, M be a left DG-module over A , and N be a right DG-module over A . Let v : A −→ k be a homogeneous k -linear retrac-tion. Define a strictly unital A ∞ -algebra structure on A by the rules m ( a ) = d ( a ), m ( a ⊗ a ) = a a , and m n = 0 for n >
2. Define a structure of a strictly unitalleft A ∞ -module over A on M by the rules l ( x ) = d ( x ), l ( a ⊗ x ) = ax , and l n = 0for i >
1, where a ∈ A and x ∈ M . Analogously, define a structure of a strictlyunital right A ∞ -module over A on N by the rules r ( y ) = d ( y ), r ( y ⊗ a ) = ya ,and r n = 0 for n >
1, where a ∈ A and y ∈ N . Then the CDG-coalgebra struc-ture Bar v ( A ) on the graded tensor coalgebra L n A + [1] ⊗ n that was defined in 6.1coincides with the CDG-coalgebra structure Bar v ( A ) constructed above, so our no-tation is consistent. The left CDG-comodule structure Bar v ( A ) ⊗ τ A,v M on thecofree graded comodule L n A + [1] ⊗ n ⊗ k M that was defined in 6.2 coincides with he left CDG-comodule structure Bar v ( A, M ). The left CDG-contramodule structureHom τ A,v (Bar v ( A ) , M ) on the free graded contramodule Hom k ( L n A + [1] ⊗ n , M ) coin-cides with the CDG-contramodule structure Cob v ( A, M ). The right CDG-comodulestructure N ⊗ τ A,v
Bar v ( A ) on the cofree graded comodule N ⊗ k L n A + [1] ⊗ n coincideswith the right CDG-comodule structure Bar v ( N, A ).A morphism of strictly unital A ∞ -algebras f : A −→ B is called strict if f n = 0 forall n >
1. An augmented strictly unital A ∞ -algebra A is a strictly unital A ∞ -algebraendowed with a morphism of strictly unital A ∞ -algebras A −→ k , where the strictlyunital A ∞ -algebra structure on k comes from its structure of DG-algebra with zerodifferential. An augmented strictly unital A ∞ -algebra is strictly augmented if theaugmentation morphism is strict. A morphism of augmented or strictly augmentedstrictly unital A ∞ -algebras is a morphism of strictly unital A ∞ -algebras forming acommutative diagram with the augmentation morphisms. The categories of aug-mented strictly unital A ∞ -algebras, strictly augmented strictly unital A ∞ -algebras,and nonunital A ∞ -algebras are equivalent. The equivalence of the latter two cat-egories is provided by the functor of formal adjoining of the strict unit, and theequivalence of the former two categories can be deduced from the equivalence be-tween the categories of DG-coalgebras C and CDG-coalgebras C endowed with aCDG-coalgebra morphism C −→ k . The DG-category of strictly unital A ∞ -modulesover an augmented strictly unital A ∞ -algebra A is equivalent to the DG-category ofnonunital A ∞ -modules over the corresponding nonunital A ∞ -algebra.7.3. Derived category of A ∞ -modules. Let A be a strictly unital A ∞ -algebraover a field k . A (not necessarily closed) morphism of strictly unital left A ∞ -modules f : L −→ M over A is called strict if one has f n = 0 and ( df ) n = 0 for all n > f n = 0 and ( df ) n = 0 for all n >
0. Strictly unital left A ∞ -modulesand strict morphisms between them form a DG-subcategory of the DG-category ofstrictly unital left A ∞ -modules and their morphisms.A closed strict morphism of strictly unital A ∞ -modules is called a strict homotopyequivalence if it is a homotopy equivalence in the DG-category of strictly unitalA ∞ -modules and strict morphisms between them. A triple K −→ L −→ M ofstrictly unital A ∞ -modules with closed strict morphisms between them is said to be exact if K −→ L −→ M is an exact triple of graded vector spaces. The total strictlyunital A ∞ -module of such an exact triple is defined in the obvious way.Any strictly unital left A ∞ -module M over A can be considered as a complexwith the differential l = p : M −→ M , since one has l = 0. A strictly unital leftA ∞ -module M is called acyclic if it is acyclic as a complex with the differential l .For any closed morphism of strictly unital left A ∞ -modules f : L −→ M the map f = f : L −→ M is a morphism of complexes with respect to l . The morphism f is called a quasi-isomorphism if f is a quasi-isomorphism of complexes. et v : A −→ k be a homogeneous k -linear retraction and C = Bar v ( A ) be thecorresponding CDG-coalgebra structure on the graded tensor coalgebra L n A + [1] ⊗ n . Theorem 1.
The following six definitions of the derived category D ( A – mod ) ofstrictly unital left A ∞ -modules over A are equivalent, i. e., lead to naturally isomor-phic (triangulated) categories: (a) the homotopy category of the DG-category of strictly unital left A ∞ -modulesover A and their morphisms; (b) the localization of the category of strictly unital left A ∞ -modules over A andtheir closed morphisms by the class of quasi-isomorphisms; (c) the localization of the category of strictly unital left A ∞ -modules over A andtheir closed morphisms by the class of strict homotopy equivalences; (d) the quotient category of the homotopy category of the DG-category of strictlyunital left A ∞ -modules over A and strict morphisms between them by the thick sub-category of acyclic A ∞ -modules; (e) the localization of the category of strictly unital left A ∞ -modules over A andtheir closed strict morphisms by the class of strict quasi-isomorphisms; (f) the quotient category of the homotopy category of the DG-category of strictlyunital left A ∞ -modules over A and strict morphisms between them by its minimaltriangulated subcategory containing all the total strictly unital A ∞ -modules of exacttriples of strictly unital A ∞ -modules with closed strict morphisms between them.The derived category D ( A – mod ) is also naturally equivalent to the following trian-gulated categories: (g) the coderived category D co ( C – comod ) of left CDG-comodules over C ; (h) the contraderived category D ctr ( C – contra ) of left CDG-contramodules over C ; (i) the absolute derived category D abs ( C – comod ) of left CDG-comodules over C ; (j) the absolute derived category D abs ( C – contra ) of left CDG-contramodules over C .Proof. The equivalence of (a-h) holds in the generality of CDG-comodules andCDG-contramodules over an arbitrary conilpotent CDG-coalgebra C . More pre-cisely, let us consider CDG-comodules over C that are cofree as graded comodules,or equivalently, CDG-contramodules over C that are free as graded contramodules,in place of strictly unital A ∞ -modules. The equivalence of (a), (g), and (h) followsfrom (the proof of) Theorem 4.4.There is a natural increasing filtration F on a conilpotent CDG-coalgebra C that was defined in 6.4, and there are induced increasing filtrations F n K = λ − ( F n C ⊗ k K ) on all CDG-comodules K over C and decreasing filtrations F n Q = π (Hom k ( C/F n − C, Q )) on all CDG-contramodules Q over C . In particular, from anyCDG-comodule C ⊗ k M that is cofree as a graded comodule and the correspond-ing CDG-contramodule Hom k ( C, M ) that is free as a graded contramodule one canrecover the complex M as M ≃ F ( C ⊗ k M ) ≃ Hom k ( C, M ) /F Hom k ( C, M ). A losed morphism of CDG-comodules C ⊗ k L −→ C ⊗ k M and the correspondingclosed morphism of CDG-contramodules Hom k ( C, L ) −→ Hom k ( C, M ) are homo-topy equivalences if and only if the corresponding morphism of complexes L −→ M is a quasi-isomorphism. Indeed, let us pass to the cones and check that a cofreeCDG-comodule C ⊗ k M is contractible if and only if the complex M is acyclic. The“only if” is clear, and “if” follows from the fact that C ⊗ k M is coacyclic when-ever M is acyclic. To check the latter, notice that the quotient CDG-comodules F n ( C ⊗ k M ) /F n − ( C ⊗ k M ) are just the tensor products of complexes of vectorspaces F n C/F n − C ⊗ k M with the trivial CDG-comodule structures.This proves the equivalence of (a) and (b), since for any DG-category DG withshifts and cones the homotopy category H ( DG ) can be also obtained by invertinghomotopy equivalences in the category of closed morphisms Z ( DG ). The equivalenceof (d) and (e) also follows from the latter result about DG-categories; and to prove theequivalence of (a) and (c) the following slightly stronger formulation of that resultis sufficient. For any DG-category DG with shifts and cones consider the class ofmorphisms of the form (id X ,
0) : X ⊕ cone(id X ) −→ X . Then by formally invertingall the morphisms in this class one obtains the homotopy category H ( DG ).A morphism of CDG-comodules f ′ : C ⊗ k L −→ C ⊗ k M and the correspondingmorphism of CDG-contramodules f ′′ : Hom k ( C, L ) −→ Hom k ( C, M ) can be calledstrict if both f ′ and df ′ as maps of graded vector spaces can be obtained by applyingthe functor C ⊗ − to certain maps L −→ M , or equivalently, both f ′′ and df ′′ asmaps of graded vector spaces can be obtained by applying the functor Hom k ( C, − )to (the same) maps L −→ M . Let w : k −→ C be the coaugmentation map; con-sider the DG-algebra U = Cob w ( C ). When C = Bar v ( A ) is the bar-constructionof a strictly unital A ∞ -algebra A , the DG-algebra U is called the envelopingDG-algebra of A . For any conilpotent CDG-coalgebra C , consider the DG-functors C ⊗ τ C,w − and Hom τ C,w ( C, − ) assigning CDG-comodules and CDG-contramodulesover C to DG-modules over U . These two DG-functors are equivalences betweenthe DG-categories of left DG-modules over U , left CDG-comodules over C thatare cofree as graded comodules with strict morphisms between them, and leftCDG-contramodules over C that are free as graded contramodules with strict mor-phisms between them. So the equivalence of (a) and (d) follows from Theorem 6.4,and the equivalence of (d) and (f) follows from Corollary 6.7.Finally, the equivalences (g) ⇐⇒ (i) and (h) ⇐⇒ (j) for C = Bar v ( A ) are providedby Theorem 4.5. (cid:3) Let A be a DG-algebra over k ; it can be considered as a strictly unital A ∞ -algebraand left DG-modules over it can be considered as strictly unital A ∞ -modules asexpained in 7.2. It follows from Theorem 6.3 that the derived category of leftDG-modules over A is equivalent to the derived category of left A ∞ -modules, soour notation D ( A – mod ) is consistent. ny strictly unital A ∞ -algebra A can be considered as a complex with the differ-ential m : A −→ A , since m = 0. For any morphism of strictly unital A ∞ -algebras f : A −→ B the map f : A −→ B is a morphism of complexes with respectto m . A morphism f of strictly unital A ∞ -algebras is called a quasi-isomorphism if f : A −→ B is a quasi-isomorphism of complexes, or equivalently, f , + : A + −→ B + is a quasi-isomorphism of complexes.Let f : A −→ B be a morphism of strictly unital A ∞ -algebras and g : Bar v ( A ) −→ Bar v ( B ) be the corresponding morphism of CDG-coalgebras. Any strictly unitalleft A ∞ -module M over B can be considered as a strictly unital left A ∞ -moduleover A ; this corresponds to the extension-of-scalarars functors E g on the levelof CDG-comodules that are cofree as graded comodules and E g on the level ofCDG-contramodules that are free as graded contramodules. Denote the inducedfunctor on derived categories by I R f : D ( B – mod ) −→ D ( A – mod ). The functor I R f has left and right adjoint functors L E f and R E f : D ( A – mod ) −→ D ( B – mod ) thatcan be constructed as the functors I R g and I R g on the level of coderived categoriesof CDG-comodules and contraderived categories of CDG-contramodules (see 5.4). Theorem 2.
The functor R f is an equivalence of triangulated categories if and onlyif a morphism f of strictly unital A ∞ -algebras is a quasi-isomorphism.Proof. The “if” part follows easily from Theorem 4.8. Both “if” and “only if” can bededuced from Theorem 1.7 in the following way. For any strictly unital A ∞ -algebra A and the corresponding CDG-coalgebra C = Bar v ( A ) with its coaugmentation w , theadjunction morphism C −→ Bar v (Cob w ( C )) corresponds to a morphism of strictlyunital A ∞ -algebras u : A −→ U ( A ) from A to its the enveloping DG-algebra U ( A )(see the proof of Theorem 1). The morphism u is a quasi-isomorphism, as one cansee by considering the increasing filtration F on A defined by the rules F A = k and F A = A , and the induced filtration on U ( A ). The functor I R u is an equivalenceof triangulated categories, as it follows from Theorems 6.3 and 6.4, or as we havejust proved. It remains to apply Theorem 1.7 to the morphism of DG-algebras U ( f ) : U ( A ) −→ U ( B ). (cid:3) Let A be a strictly unital A ∞ -algebra and C = Bar v ( A ) be the correspondingCDG-coalgebra. All the above results about strictly unital left A ∞ -modules over A apply to strictly unital right A ∞ -modules as well, since one can pass to the op-posite CDG-coalgebra C op as defined in 4.7. In particular, the derived category ofstrictly unital right A ∞ -modules D ( mod – A ) is defined and naturally equivalent tothe coderived category D co ( comod – C ).The functor Tor A : D ( mod – A ) × D ( A – mod ) −→ k – vect gr can be constructedeither by restricting the functor of cotensor product (cid:3) C : Hot ( comod – C ) × Hot ( C – comod ) −→ Hot ( k – vect ) to the Cartesian product of the homotopy categoriesof CDG-comodules that are cofree as graded comodules, or by restricting the functor f contratensor product ⊙ C : Hot ( comod – C ) × Hot ( C – contra ) −→ Hot ( k – vect ) to theCartesian product of the homotopy categories of CDG-comodules that are cofree asgraded comodules and CDG-contramodules that are free as graded contramodules.The functors one obtains in these two ways are naturally isomorphic by the resultof 5.3. This definition of the functor Tor A agrees with the definition of functor Tor A for DG-algebras A by Theorem 6.9.1.The functor Ext A = Hom D ( A – mod ) : D ( A – mod ) op × D ( A – mod ) −→ k – vect gr can becomputed in three ways. One can either restrict the functor Hom C : Hot ( C – comod ) op × Hot ( C – comod ) −→ Hot ( k – vect ) to the Cartesian product of the homotopy cate-gories of CDG-comodules that are cofree as graded comodules, or restrict the functorHom C : Hot ( C – contra ) op × Hot ( C – contra ) −→ Hot ( k – vect ) to the Cartesian prod-uct of the homotopy categories of CDG-contramodules that are free as graded con-tramodules, or restrict the functor Cohom C : Hot ( C – comod ) op × Hot ( C – contra ) −→ Hot ( k – vect ) to the Cartesian product of the homotopy categories of CDG-comodulesthat are cofree as graded comodules and CDG-contramodules that are free as gradedcontramodules. The functors one obtains in these three ways are naturally isomorphicby the result of 5.3, are isomorphic to the functor Hom D ( A – mod ) by Theorem 1 above,and agree with the functor Ext A for DG-algebras A by Theorem 6.3 or Theorem 6.9.1. Remark.
One can define a nonunital curved A ∞ -algebra A as a structure of notnecessarily coaugmented DG-coalgebra on L n A [1] ⊗ n ; such a structure is given by asequence of linear maps m n : A ⊗ n −→ A , n = 0, 1, . . . , where m : k −→ A maybe a nonzero map (corresponding to the curvature element of A ). Any morphismof DG-coalgebras f : L n A [1] ⊗ n −→ B [1] ⊗ n preserves the coaugmentations of thegraded tensor coalgebras, though, so a morphism of nonunital curved A ∞ -algebras f : A −→ B is given by a sequence of linear maps f n : A ⊗ n −→ B , n = 1, 2, . . . Allthe definitions of 7.1–7.2 can be generalized straightforwardly to the curved situation,and all the results of 7.2 hold in this case. However, this theory is largely trivial. Forany strictly unital curved A ∞ -algebra A with m = 0, every object of the DG-categoryof strictly unital curved A ∞ -modules over A is contractible. In particular, the sameapplies to nonunital curved A ∞ -modules over a nonunital curved A ∞ -algebra. Twocases have to be considered separately, the case when the image of m coincideswith k · A ⊂ A , and the case when m (1) and 1 A are linearly independent. Theformer case cannot occur in the Z -graded situation for dimension reasons, but in the Z / C = L n A + [1] ⊗ n is compatible with the coaugmentation w : k ≃ A + [1] ⊗ −→ C , butthe curvature linear function h : C −→ k is not, i. e., h ◦ w = 0. Let C ⊗ k M bea left CDG-comodule over C that is cofree as a graded comodule and F n C ⊗ k M be its natural increasing filtration induced by the natural increasing filtration F ofthe conilpotent graded coalgebra C . Then the filtrations F on both C and C ⊗ k M are preserved by the differentials, since the differential on C is compatible with the oaugmentation. The induced differential l : M −→ M on M = F C ⊗ k M has thesquare equal to a nonzero constant from k times the identity endomorphism of M .The CDG-comodule ( M, l ) over the CDG-coalgebra F C is clearly contractible; let t be its contracting homotopy. Set t = id ⊗ t : C ⊗ k M −→ C ⊗ k M ; then t is a nonclosed endomorphism of M of degree − d ( t ) = dt + td is invertible, hence C ⊗ k M is contractible. This argument is applicable toany conilpotent graded coalgebra C . In the case when m (1) and 1 A are linearlyindependent, the theory trivializes even further. The author learned the idea ofthe following arguments from M. Kontsevich. All strictly unital curved A ∞ -algebrastructures with m (1) and 1 A linearly independent on a given graded vector space A are isomorphic, and all structures of a strictly unital curved A ∞ -module over A on agiven graded vector space M are isomorphic. Indeed, consider the component of thetensor degree n = 1 of the differential on C = L n A + [1] ⊗ n ; it is determined by m .This differential makes L n A + [1] ⊗ n into a complex, and this complex is acyclic.Taking this fact into account, one can first find a CDG-coalgebra isomorphism of theform (id , a ), a : C −→ k between a given CDG-coalgebra structure on C and a certainCDG-coalgebra structure with h = 0, i. e., a DG-coalgebra structure. One proceedsstep by step, killing the component h n : A ⊗ n + −→ k of the linear function h using alinear function a with the only component a n +1 : A ⊗ n +1+ −→ k . Having obtained aDG-coalgebra structure on C , one subsequently kills all the components m n : A ⊗ n + −→ A of the differential d with n > f of C with the only nonzero components f = id A + and f n +1 : A ⊗ n +1+ −→ A + . Analogouslyone shows that any DG-comodule over C that is cofree as a graded comodule isisomorphic to a direct sum of shifts of the DG-comodule C . Since C is acyclic,such DG-comodules are clearly contractible. Consequently, the coderived category D co ( C – comod ) vanishes. Alternatively, one could consider the DG-category of strictlyunital curved A ∞ -modules over a strictly unital curved A ∞ -algebra A with strictmorphisms between the curved A ∞ -modules. This DG-category is equivalent to theDG-category of CDG-modules over the CDG-coalgebra U = Cob w ( C ). Its homotopycategory Hot ( U – mod ) can well be nonzero, but the corresponding absolute derivedcategory D abs ( U – mod ) is zero by Theorem 6.7. So in the category of (strictly unital ornonunital) curved A ∞ -algebras over a field there are too few and too many morphismsat the same time: there are no “change-of-connection” morphisms, and in particularno morphisms corresponding to nonstrict morphisms of CDG-algebras, and still thereare enough morphisms to trivialize the theory in almost all cases. One way out of thispredicament is to restrict oneself to the curved A ∞ -algebras (modules, morphisms)for each of which the structure maps m n , l n , f n vanish for n large enough. For suchcurved A ∞ -algebras A , it actually makes sense to consider morphisms f : A −→ B with nonvanishing change-of-connection components f : k −→ B . This theory can beinterpreted in terms of the topological tensor coalgebras Q n A [1] ⊗ n (cf. Remark 7.6). nother solution would be to consider curved A ∞ -algebras over the ring of formalpower series k [[ t ]] and require the curvature and change-of-connection elements to bedivisible by t .7.4. Noncounital curved A ∞ -coalgebras. Let C be a graded vector space overa field k . Consider the graded tensor algebra (free associative graded algebra) L ∞ n =0 C [ − ⊗ n generated by the graded vector space C [ − noncounital curved A ∞ -coalgebra structure on C is, by the definition, a DG-algebra structure on L n C [ − ⊗ n , i. e., an odd derivation d of degree 1 on L n C [ − ⊗ n such that d = 0. Since a derivation of L n C [ − ⊗ n is uniquely determined by its restric-tion to C [ − ≃ C [ − ⊗ ⊂ L n C [ − ⊗ n , a noncounital curved A ∞ -coalgebrastructure on C can be considered as a sequence of linear maps µ n : C −→ C ⊗ n , n = 0, 1, . . . of degree 2 − n . More precisely, define the maps µ n by the for-mula d ( c ) = P ∞ n =0 ( − n + P nj =1 ( n − j )( | µ n,j ( c ) | +1) µ n, ( c ) ⊗ · · · ⊗ µ n,n ( c ), where c ∈ C and µ n ( c ) = µ n, ( c ) ⊗ · · · ⊗ µ n,n ( c ) is a symbolic notation for the tensor µ n ( c ) ∈ C ⊗ n . A convergence condition must be satisfied: for any c ∈ C one must have µ n ( c ) = 0 forall but a finite number of the degrees n . Furthermore, the sequence of maps µ n mustsatisfy a sequence of quadratic equations corresponding to the equation d = 0 onthe derivation d .A morphism of noncounital curved A ∞ -coalgebras f : C −→ D over a field k is,by the definition, a morphism of DG-algebras L n C [ − ⊗ n −→ L n D [ − ⊗ n over k .Since the graded algebra morphism from a graded tensor algebra L n C [ − ⊗ n isdetermined by its restriction to C [ − ≃ C [ − ⊗ ⊂ L n C [ − ⊗ n , a morphism ofnoncounital curved A ∞ -coalgebras f : C −→ D can be considered as a sequence oflinear maps f n : C −→ D ⊗ n , n = 0, 1, . . . of degree 1 − n . More precisely, define themaps f n by the formula f ( c ) = P ∞ n =0 ( − n − P nj =1 ( n − j )( | f n,j ( c ) | +1) f n, ( c ) ⊗· · ·⊗ f n,n ( c ),where c ∈ C and f n ( c ) = f n, ( c ) ⊗ · · · ⊗ f n,n ( c ) ∈ D ⊗ n . A convergence condition mustbe satisfied: for any c ∈ C one must have f n ( c ) = 0 for all but a finite numberof the degrees n . Furthermore, the sequence of maps f n must satisfy a sequence ofpolynomial equations corresponding to the equation d ◦ f = f ◦ d on the morphism f .Let C be a noncounital curved A ∞ -coalgebra over a field k and M be a gradedvector space over k . A structure of noncounital left curved A ∞ -comodule over C on M is, by the definition, a structure of DG-module over the DG-algebra L n C [ − ⊗ n onthe free graded left module L n C [ − ⊗ n ⊗ k M over the graded algebra L n C [ − ⊗ n .Analogously, a structure of noncounital right curved A ∞ -comodule over C on a gradedvector space N is defined as a structure of DG-module over L n C [ − ⊗ n on thefree graded right module N ⊗ k L n C [ − ⊗ n . Since a derivation of a free gradedmodule L n C [ − ⊗ n ⊗ k M compatible with a given derivation of the graded al-gebra L n C [ − ⊗ n is determined by its restriction to the subspace of generators M ≃ C [ − ⊗ ⊗ k M ⊂ L n C [ − ⊗ n ⊗ k M , a noncounital left curved A ∞ -comodulestructure on M can be considered as a sequence of linear maps λ n : M −→ C ⊗ n ⊗ k M , = 0, 1, . . . of degree 1 − n . More precisely, define the maps λ n by the formula d ( x ) = P ∞ n =0 ( − n + P − j = − n ( j +1)( | λ n,j ( x ) | +1) λ n, − n ( x ) ⊗ · · · ⊗ λ n, − ( x ) ⊗ λ n, ( x ), where x ∈ M and λ n ( x ) = λ n, − n ( x ) ⊗ · · · ⊗ λ n, − ( x ) ⊗ λ n, ( x ) ∈ C ⊗ n ⊗ k M . A conver-gence condition must be satisfied: for any x ∈ M one must have λ n ( x ) = 0 forall but a finite number of the degrees n . Furthermore, the sequence of maps λ n must satisfy a sequence of nonhomogeneous quadratic equations corresponding tothe equation d = 0 on the derivation d on L n C [ − ⊗ n ⊗ k M . Analogously,a noncounital right curved A ∞ -comodule structure on N can be considered as asequence of linear maps ρ n : N −→ N ⊗ k C ⊗ n defined by the formula d ( y ) = P ∞ n =0 ( − n | ρ n, ( y ) | + P nj =1 ( n − j )( | ρ n,j ( y ) | +1) ρ n, ( y ) ⊗ ρ n, ( y ) ⊗ · · · ⊗ ρ n,n ( y ), where y ∈ N and ρ n ( y ) = ρ n, ( y ) ⊗ ρ n, ( y ) ⊗ · · · ⊗ ρ n,n ( y ) ∈ N ⊗ k C ⊗ n .The complex of morphisms between noncounital left curved A ∞ -comodules L and M over a noncounital curved A ∞ -coalgebra C is, by the definition, the complexof morphisms between left DG-modules L n C [ − ⊗ n ⊗ k L and L n C [ − ⊗ n ⊗ k M over the DG-algebra L n C [ − ⊗ n . Analogously, the complex of morphisms be-tween noncounital right curved A ∞ -comodules R and N over C is, by the defini-tion, the complex of morphisms between right DG-modules R ⊗ k L n C [ − ⊗ n and N ⊗ k L n C [ − ⊗ n over the DG-algebra L n C [ − ⊗ n . A morphism of noncounitalleft curved A ∞ -comodules f : L −→ M of degree i is the same that a sequence oflinear maps f n : L −→ C ⊗ n ⊗ k M , n = 0, 1, . . . of degree i − n satisfying theconvergence condition: for any x ∈ L one must have f n ( x ) = 0 for all but a fi-nite number of the degrees n . More precisely, define the maps f n by the formula f ( x ) = P ∞ n =0 ( − n + P − j = − n ( j +1)( | f n,j ( x ) | +1) f n, − n ( x ) ⊗ · · · ⊗ f n, − n ( x ) ⊗ f n, ( x ), where x ∈ L and f n ( x ) = f n, − n ( x ) ⊗ · · · ⊗ f n, − ( x ) ⊗ f n, ( x ) ∈ C ⊗ n ⊗ k M . Any sequence oflinear maps f n satisfying the convergence condition corresponds to a (not necessarilyclosed) morphism of noncounital curved A ∞ -comodules f . Analogously, a morphismof noncounital right curved A ∞ -comodules f : R −→ N of degree i is the same thata sequence of linear maps f n : R −→ N ⊗ k C ⊗ n , n = 0, 1, . . . of degree i − n satis-fying the convergence condition. More precisely, define the maps f n by the formula f ( y ) = P ∞ n =0 ( − n | f n, ( y ) | + P nj =1 ( n − j )( | f n,j ( y ) | +1) f n, ( y ) ⊗ f n, ( y ) ⊗ · · · ⊗ f n,n ( y ), where y ∈ R and f n ( y ) = f n, ( y ) ⊗ f n, ( y ) ⊗ · · · ⊗ f n,n ( y ) ∈ N ⊗ k C ⊗ n .Let C be a noncounital curved A ∞ -coalgebra over a field k and P be a graded vec-tor space over k . A structure of noncounital left curved A ∞ -contramodule over C on P is, by the definition, a structure of DG-module over the DG-algebra L n C [ − ⊗ n on the cofree graded left module Hom k ( L n C [ − ⊗ n , P ) over the graded algebra L n C [ − ⊗ n . The action of L n C [ − ⊗ n in Hom k ( L n C [ − ⊗ n , P ) is induced by theright action of L n C [ − ⊗ n in itself as explained in 1.5 and 1.7. Since a derivation ofa cofree graded module Hom k ( L n C [ − ⊗ n , P ) compatible with a given derivation ofthe graded algebra L n C [ − ⊗ n is determined by its composition with the projection om k ( L n C [ − ⊗ n , P ) −→ P induced by the unit map k −→ L n C [ − ⊗ n , a non-counital left curved A ∞ -contramodule structure on P can be considered as a linearmap π : Q ∞ n =0 Hom k ( C ⊗ n , P )[ n − −→ P of degree 0. More precisely, define the map π by the rule that the image of the element d ( g ) under the projection to P equals π (( g n ) ∞ n =0 ), where a map g : L n C [ − ⊗ n −→ P and a sequence of maps g n : C ⊗ n −→ P are related by the formula g ( c ⊗· · ·⊗ c n ) = ( − n | g n | + P nj =1 ( n − j )( | c j | +1) g n ( c ⊗· · ·⊗ c n )for c j ∈ C . The map π must satisfy a system of nonhomogeneous quadratic equationscorresponding to the equation d = 0 on the derivation d on Hom k ( L n C [ − ⊗ n , P ).The complex of morphisms between noncounital left curved A ∞ -contramodules P and Q over a noncounital curved A ∞ -coalgebra C is, by the definition, thecomplex of morphisms between left DG-modules Hom k ( L n C [ − ⊗ n , P ) andHom k ( L n C [ − ⊗ n , Q ) over the DG-algebra L n C [ − ⊗ n . A morphism of non-counital left curved A ∞ -contramodules f : P −→ Q of degree i is the same that alinear map f ⊓ : Q ∞ n =0 Hom k ( C ⊗ n , P )[ n ] −→ Q of degree i . More precisely, definethe map f ⊓ by the rule that the image of the element f ( g ) under the projection to Q equals f ⊓ (( g n ) ∞ n =0 ), where a map g : L n C [ − ⊗ n −→ P and a sequence of maps g n : C ⊗ n −→ P are related by the above formula. Any linear map f ⊓ corresponds toa (not necessarily closed) morphism of noncounital curved A ∞ -contramodules f .7.5. Strictly counital curved A ∞ -coalgebras. Let C be a noncounital curvedA ∞ -coalgebra over a field k . A homogeneous linear function ε : C −→ k of de-gree 0 is called a strict counit if one has ε ( µ , ( c )) µ , ( c ) = c = ε ( µ , ( c )) µ , ( c )and ε ( µ n,j ( c )) µ n, ( c ) ⊗ · · · ⊗ µ n,j − ( c ) ⊗ µ n,j +1 ( c ) ⊗ · · · ⊗ µ n,n ( c ) = 0 for all c ∈ C ,1 j n , and n = 2. A strict counit is unique if it exists. A strictly couni-tal curved A ∞ -coalgebra is a noncounital curved A ∞ -coalgebra that admits a strictcounit. A morphism of strictly counital curved A ∞ -coalgebras f : C −→ D isa morphism of noncounital curved A ∞ -coalgebras such that ε D ◦ f = ε C and ε ( f n,j ( c )) f n, ( c ) ⊗· · · ⊗ f n,j − ( c ) ⊗ f n,j +1 ( c ) ⊗· · · ⊗ f n,n ( c ) = 0 for all c ∈ C , 1 j n ,and n >
1. Notice that for a strictly counital curved A ∞ -coalgebra C one has ε C = 0if and only if C = 0. We will assume our strictly counital curved A ∞ -coalgebras tohave nonzero counits.A strictly counital left curved A ∞ -comodule M over a strictly counital curvedA ∞ -coalgebra C is a noncounital left curved A ∞ -comodule such that ε ( λ , − ( x )) λ , ( x )= x and ε ( λ n,j ( x )) λ n, − n ( x ) ⊗ · · · ⊗ λ n,j − ( x ) ⊗ λ n,j +1 ( x ) ⊗ · · · ⊗ λ n, − ( x ) ⊗ λ n, ( x ) = 0for all x ∈ M , − n j −
1, and n >
1. Analogously, a strictly counitalright curved A ∞ -comodule N over C is a noncounital right curved A ∞ -comodulesuch that ε ( ρ , ( y )) ρ , ( y ) = y and ε ( ρ n,j ( y )) ρ n, ( y ) ⊗ ρ n, ( y ) ⊗ · · · ⊗ ρ n,j − ( y ) ⊗ ρ n,j +1 ( y ) ⊗ · · · ⊗ ρ n,n ( y ) = 0 for all y ∈ N , 1 j n , and n >
1. Finally, a strictly counital left curved A ∞ -contramodule P over C is a noncounital left curvedA ∞ -contramodule satisfying the following condition. For any sequence of linear aps g n : C ⊗ n −→ P , n = 0, 1, . . . of degree s + n − g ′ n,j : C ⊗ n − −→ P , 1 j n such that g n ( c ⊗ · · · ⊗ c n ) = P nj =1 ε ( c j ) g ′ n,j ( c ⊗ · · · ⊗ c j − ⊗ c j +1 ⊗ · · · ⊗ c n ) for all n and c t ∈ C the equation π (( g n ) ∞ n =0 ) = g ′ , (1) must hold in P s .The complex of morphisms between strictly counital left curved A ∞ -comodules L and M over a strictly counital left curved A ∞ -coalgebra C is the subcomplex of thecomplex of morphisms between L and M as noncounital A ∞ -comodules consisting ofall morphisms f : L −→ M such that ε ( f n,j ( x )) f n, − n ( x ) ⊗ · · · ⊗ f n,j − ( x ) ⊗ f n,j +1 ( x ) ⊗· · · ⊗ f n, − ( x ) ⊗ f n, ( x ) = 0 for all x ∈ L , − n j −
1, and n >
0. Analogously,the complex of morphisms between strictly counital right curved A ∞ -comodules R and N over C is the subcomplex of the complex of morphisms between R and N as noncounital A ∞ -comodules consisting of all morphisms f : R −→ N such that ε ( f n,j ( y )) f n, ( y ) ⊗ f n, ( y ) ⊗ · · · ⊗ f n,j − ( y ) ⊗ f n,j +1 ( y ) ⊗ · · · ⊗ f n,n ( y ) = 0 for all y ∈ R ,1 j n , and n >
0. Finally, the complex of morphisms between strictly counitalleft curved A ∞ -contramodules P and Q over C is the subcomplex of the complexof morphisms between P and Q as noncounital A ∞ -contramodules consisting of allmorphisms f : P −→ Q satisfying the following condition. For any sequence oflinear maps g n : C ⊗ n −→ P , n = 0, 1, . . . of degree s + n − i for which thereexists a double sequence of linear maps g ′ n,j : C ⊗ j − −→ P , 1 j n such that g n ( c ⊗ · · · ⊗ c n ) = P nj =1 ε ( c j ) g ′ n,j ( c ⊗ · · · ⊗ c j − ⊗ c j +1 ⊗ · · · ⊗ c n ) for all n and c t ∈ C the equation f ⊓ (( g n ) ∞ n =0 ) = 0 must hold in Q s , where i is the degree of f .Let C be a noncounital curved A ∞ -coalgebra and ε C : C −→ k be a homoge-neous linear function of degree 0. Set C + = ker ε . Then the graded tensor algebra L n C + [ − ⊗ n is a subalgebra of the tensor algebra L n C [ − ⊗ n . Denote by K C thecokernel of the embedding L n C + [ − ⊗ n −→ L n C [ − ⊗ n and by κ C ∈ K C the ele-ment of C/C + [ − ⊂ K C for which ε ( κ C ) = 1. Let θ C ∈ L n C [ − ⊗ n be any elementof degree 1 whose image in K C is equal to κ C . Then the linear function ε C : C −→ k is a strict counit if and only if the odd derivation d ′ ( a ) = d ( a ) + [ θ C , a ] of degree 1on the tensor algebra L n C [ − ⊗ n preserves the subalgebra L n C + [ − ⊗ n and theelement h = d ( θ C ) + θ C belongs to L n C + [ − ⊗ n . This condition does not dependon the choice of θ C . For strictly counital curved A ∞ -coalgebras C and D , a mor-phism of noncounital curved A ∞ -coalgebras f : C −→ D is a morphism of strictlycounital curved A ∞ -coalgebras if and only if f ( L n C + [ − ⊗ n ) ⊂ L n D + [ − ⊗ n and f ( κ C ) = κ D .Let C be a strictly counital curved A ∞ -coalgebra and M be a noncounital leftcurved A ∞ -comodule over C . Then M is a strictly counital curved A ∞ -comoduleif and only if the odd derivation d ′ ( z ) = d ( z ) + θ C z of degree 1 on the free mod-ule L n C [ − ⊗ n ⊗ k M compatible with the derivation d ′ of the algebra L n C [ − ⊗ n preserves the subspace L n C + [ − ⊗ n ⊗ k M ⊂ L n C [ − ⊗ n ⊗ k M . Analogously,a noncounital right curved A ∞ -comodule N over C is a strictly counital curved ∞ -comodule if and only if the odd derivation d ′ ( z ) = d ( z ) − ( − | z | zθ C of degree 1on the free module N ⊗ k L n C [ − ⊗ n compatible with the derivation d ′ of the algebra L n C [ − ⊗ n preserves the subspace N ⊗ k L n C + [ − ⊗ n ⊂ N ⊗ k L n C [ − ⊗ n . Finally,a noncounital left curved A ∞ -contramodule P over C is a strictly counital curvedA ∞ -contramodule if and only if the odd derivation d ′ ( q ) = d ( q ) + θ C q of degree 1on the cofree module Hom k ( L n C [ − ⊗ n , P ) compatible with the derivation d ′ of thealgebra L n C [ − ⊗ n preserves the subspace Hom k ( K C , P ) ⊂ Hom k ( L n C [ − ⊗ n , P ).For strictly counital left curved A ∞ -comodules L and M over a strictly couni-tal curved A ∞ -coalgebra C , a (not necessarily closed) morphism of noncounitalA ∞ -comodules f : L −→ M is a morphism of strictly counital A ∞ -comodules if andonly if one has f ( L n C + [ − ⊗ n ⊗ k L ) ⊂ L n C + [ − ⊗ n ⊗ k M . Analogously, for strictlycounital right curved A ∞ -comodules R and N over C , a (not necessarily closed) mor-phism of noncounital A ∞ -comodules f : R −→ N is a morphism of strictly counitalA ∞ -comodules if and only if one has f ( R ⊗ k L n C + [ − ⊗ n ) ⊂ N ⊗ k L n C + [ − ⊗ n .Finally, for strictly counital left curved A ∞ -contramodules P and Q over C , a (notnecessarily closed) morphism of noncounital A ∞ -contramodules f : P −→ Q is a mor-phism of strictly counital A ∞ -contramodules if and only if one has f (Hom k ( K C , P )) ⊂ Hom k ( K C , Q ).Let C be a strictly counital curved A ∞ -coalgebra. Choose a homogeneous k -linearsection w : k −→ C of the strict counit map ε : C −→ k . Define the element θ C ∈ L n C [ − ⊗ n as θ C = w (1) ∈ C [ − ⊂ L n C [ − ⊗ n . Then θ C is an element ofdegree 1 whose image in K C is equal to κ C . Let d : L n C + [ − ⊗ n −→ L n C + [ − ⊗ n be the restriction of the odd derivation d ′ of L n C [ − ⊗ n defined by the above for-mula, and let h ∈ L C + [ − ⊗ n be the element defined above. Then Cob w ( C ) =( L n C + [ − ⊗ n , d, h ) is a CDG-algebra. The CDG-algebra Cob w ( C ) is called the cobar-construction of a strictly counital curved A ∞ -coalgebra C .Let f : C −→ D be a morphism of strictly counital curved A ∞ -coalgebras. Let w : k −→ C and w : k −→ D be homogeneous k -linear sections, and let θ C and θ D bethe corresponding elements of degree 1 in the graded tensor algebras. The morphismof tensor algebras f : L n C [ − ⊗ n −→ L n D [ − ⊗ n induces a morphism of gradedtensor algebras L n C + [ − ⊗ n −→ L n D + [ − ⊗ n , which we will denote also by f .The element η f = f ( θ C ) − θ D ∈ L n D [ − ⊗ n has a zero image in K D , so it belongs to L n D + [ − ⊗ n . Then the pair ( f, η f ) is a morphism of CDG-algebras Cob w ( C ) −→ Cob w ( D ). Thus the cobar-construction C Cob w ( C ) is a functor from the categoryof strictly counital curved A ∞ -coalgebras with nonzero counits to the category ofCDG-algebras whose underlying graded algebras are graded tensor algebras. Onecan easily see that this functor is an equivalence of categories. Alternatively, one canuse any element θ C of degree 1 whose image in K C is equal to κ C in the constructionof this equivalence of categories. et C be a strictly counital curved A ∞ -coalgebra, w : k −→ C be a homogeneous k -linear section, and θ C ∈ L n C [ − ⊗ n be the corresponding element of degree 1.Let M be a strictly counital curved A ∞ -comodule over C . Set d : L n C + [ − ⊗ n ⊗ k M −→ L n C + [ − ⊗ n ⊗ k M to be the restriction of the differential d ′ on L n C [ − ⊗ n defined by the above formula. Then Cob w ( C, M ) = ( L n C + [ − ⊗ n ⊗ k M, d ) is a leftCDG-module over the CDG-algebra Cob w ( C ). Analogously, for a strictly counitalcurved A ∞ -comodule N over C set d : N ⊗ k L n C + [ − ⊗ n −→ N ⊗ k L n C + [ − ⊗ n to be the restriction of the differential d ′ on N ⊗ k L n C [ − ⊗ n defined above. ThenCob w ( N, C ) = ( N ⊗ k L n C + [ − ⊗ n , d ) is a right CDG-module over the CDG-algebraCob w ( C ). Finally, let P be a strictly counital curved A ∞ -contramodule over C .Set d : Hom k ( L n C + [ − ⊗ n , P ) −→ Hom k ( L n C + [ − ⊗ n , P ) to be the map inducedby the differential d ′ on Hom k ( L n C [ − ⊗ n , P ) defined above. Then Bar w ( C, P ) =(Hom k ( L n C + [ − ⊗ n , P ) , d ) is a left CDG-module over the CDG-algebra Cob w ( C ).To a (not necessarily closed) morphism of strictly counital left curved A ∞ -comod-ules f : L −→ M over C one can assign the induced map L n C + [ − ⊗ n ⊗ k L −→ L n C + [ − ⊗ n ⊗ k M . So we obtain the DG-functor M Cob w ( C, M ), which is anequivalence between the DG-category of strictly counital left curved A ∞ -comodulesover C and the DG-category of left CDG-modules over Cob w ( C ) that are free asgraded modules. Analogously, to a (not necessarily closed) morphism of strictlycounital right curved A ∞ -comodules f : R −→ N over C one can assign the inducedmap R ⊗ k L n C + [ − ⊗ n −→ N ⊗ k L n C + [ − ⊗ n . So we obtain the DG-functor N Cob w ( N, C ), which is an equivalence between the DG-categories of strictly counitalright curved A ∞ -comodules over C and right CDG-modules over Cob w ( C ) that arefree as graded modules. Finally, to a (not necessarily closed) morphism of strictlycounital left curved A ∞ -contramodules f : P −→ Q over C one can assign the inducedmap Hom k ( L n C ⊗ n + , P ) −→ Hom k ( L n C ⊗ n + , Q ). So we obtain the DG-functor P Bar w ( C, P ), which is an equivalence between the DG-category of strictly counital leftcurved A ∞ -contramodules over C and the DG-category of left CDG-modules overCob w ( C ) that are cofree as graded modules.Now let C be a CDG-coalgebra with a nonzero counit, M be a left CDG-comoduleover C , N be a right CDG-comodule over C , and P be a left CDG-contramoduleover C . Let w : k −→ C be a homogeneous k -linear section. Define a strictly counitalcurved A ∞ -coalgebra structure on C by the rules µ ( c ) = h ( c ), µ ( c ) = d ( c ), µ ( c ) = c (1) ⊗ c (2) , and µ n ( c ) = 0 for n >
2. Define a structure of strictly counital left curvedA ∞ -comodule over C on M by the rules λ ( x ) = d ( x ), λ ( x ) = x ( − ⊗ x (0) , and λ n ( x ) = 0 for n >
1, where x ∈ M . Define a structure of strictly counital rightcurved A ∞ -comodule over C on N by the rules ρ ( y ) = d ( y ), ρ ( y ) = y (0) ⊗ y (1) ,and ρ n ( x ) = 0 for n >
1, where y ∈ N . Finally, define a structure of strictlycounital left curved A ∞ -contramodule over C on P by the rule π (( g n ) ∞ n =0 ) = d ( g ) + π P ( g ), where d : P −→ P is the differential on P and π P : Hom k ( C, P ) −→ P is the ontraaction map. Then the CDG-algebra structure Cob w ( C ) on the graded tensoralgebra L n C [ − ⊗ n that was defined in 6.1 coincides with the CDG-algebra structureCob w ( C ) constructed above, so our notation is consistent. The left CDG-modulestructure Cob w ( C ) ⊗ τ C,w M on the free graded module L n C [ − ⊗ n ⊗ k M that wasdefined in 6.2 coincides with the left CDG-module structure Cob w ( C, M ). The rightCDG-module structure N ⊗ τ C,w
Cob w ( C ) on the free graded module N ⊗ k L n C [ − ⊗ n coincides with the right CDG-module structure Cob w ( N, C ). The left CDG-modulestructure Hom τ C,w (Cob w ( C ) , P ) on the cofree graded module Hom k ( L n C [ − ⊗ n , P )coincides with the left CDG-module structure Bar w ( C, P ).A morphism of strictly counital curved A ∞ -coalgebras f : C −→ D is called strict if f n = 0 for all n = 1. A coaugmented strictly counital curved A ∞ -coalgebra C isa strictly counital curved A ∞ -coalgebra endowed with a morphism of strictly couni-tal curved A ∞ -coalgebras k −→ C , where the strictly counital curved A ∞ -coalgebrastructure on k comes from its structure of CDG-coalgebra with zero differential andcurvature linear function. A coaugmented strictly counital curved A ∞ -coalgebrais strictly coaugmented if the coaugmentation morphism is strict. A morphism ofcoaugmented or strictly coaugmented strictly counital curved A ∞ -coalgebras is amorphism of strictly counital curved A ∞ -coalgebras forming a commutative diagramwith the coaugmentation morphisms. The categories of coaugmented strictly counitalcurved A ∞ -coalgebras, strictly coaugmented strictly counital curved A ∞ -coalgebras,and noncounital curved A ∞ -coalgebras are equivalent. The DG-category of strictlycounital curved A ∞ -comodules or A ∞ -contramodules over a coaugmented strictlycounital curved A ∞ -coalgebra C is equivalent to the DG-category of noncouni-tal curved A ∞ -comodules or A ∞ -contramodules over the corresponding noncouni-tal curved A ∞ -coalgebra C . If C is a strictly coaugmented strictly counital curvedA ∞ -coalgebra and w : k −→ C is the coaugmentation map, then the CDG-algebraCob w ( C ) is in fact a DG-algebra.7.6. Coderived category of curved A ∞ -comodules and contraderived cate-gory of curved A ∞ -contramodules. Let C be a strictly counital curved A ∞ -coal-gebra over a field k . Let w : k −→ C be a homogeneous k -linear section and B = Cob w ( C ) be the corresponding CDG-algebra structure on L n C + [ − ⊗ n .The coderived category D co ( C – comod ) of strictly counital left curved A ∞ -comodules over C is defined as the homotopy category of the DG-category of strictly counitalleft curved A ∞ -comodules over C . The coderived category D co ( comod – C ) of strictlycounital right curved A ∞ -comodules over C is defined in the analogous way. The con-traderived category D ctr ( C – contra ) of strictly counital right curved A ∞ -contramodules over C is defined as the homotopy category of the DG-category of strictly counitalleft curved A ∞ -contramodules over C . Theorem.
The following five triangulated categories are naturally equivalent: a) the coderived category D co ( C – comod ) ; (b) the contraderived category D ctr ( C – contra ) ; (c) the coderived category D co ( B – mod ) ; (d) the contraderived category D ctr ( B – mod ) ; (e) the absolute derived category D abs ( B – mod ) .Proof. The isomorphism of triangulated categories (c–e) is provided by Theo-rem 3.6(a), and the equivalence of triangulated categories (a), (b), and (e) is theassertion of Theorem 3.6(b) with projective and injective graded modules replacedby free and cofree ones. It suffices to find for any left CDG-module M over B aclosed injection from M to a CDG-module J such that both J and J/M are cofree asgraded B -modules, and a closed surjection onto M from a CDG-module F such thatboth CDG-modules M and ker( F → M ) are free as graded B -modules. This can beeasily accomplished with either of the constructions of Theorem 3.6 or Theorem 4.4.One only has to notice that for any graded module M over a graded tensor algebra B the kernel of the map B ⊗ k M −→ M is a free graded B -module and the cokernelof the map M −→ Hom k ( B, M ) is a cofree graded B -module. (cid:3) Let C be a CDG-coalgebra over k ; it can be considered as a strictly counitalcurved A ∞ -coalgebra, and CDG-comodules and CDG-contramodules over it can beconsidered as strictly counital curved A ∞ -comodules and A ∞ -contramodules as ex-plained in 7.5. It follows from Theorem 6.7 that the coderived category of leftCDG-comodules over C is equivalent to the coderived category of strictly counitalleft curved A ∞ -comodules and the contraderived category of left CDG-contramodulesover C is equivalent to the contraderived category of strictly counital left curvedA ∞ -contramodules, so our notation is consistent. Thus the above Theorem providesthe comodule-contramodule correspondence for strictly counital curved A ∞ -coalgebras .By Theorem 6.7(c), the comodule-contramodule correspondence functors in theCDG-coalgebra case agree with the comodule-contramodule correspondence functorswe have construced in the strictly counital curved A ∞ -coalgebra case.For the definition of a conilpotent curved A ∞ -coalgebra and the curved A ∞ -coal-gebra analogues of some assertions of Theorem 7.3, see Remark 9.4.The functor Cotor C : D co ( comod – C ) × D co ( C – comod ) −→ k – vect gr is constructedby restricting the functor of tensor product ⊗ B : Hot ( mod – B ) × Hot ( B – mod ) −→ Hot ( k – vect ) to the Cartesian product of the homotopy categories of CDG-mod-ules that are free as graded modules. The functor Coext C : D co ( C – comod ) op × D ctr ( C – contra ) −→ k – vect gr is constructed by restricting the functor of homomor-phisms Hom B : Hot ( B – mod ) op × Hot ( B – mod ) −→ Hot ( k – vect ) to the Cartesian prod-uct of the homotopy category of CDG-modules that are free as graded modules and he homotopy category of CDG-modules that are cofree as graded modules. The func-tor Ctrtor C : D co ( comod – C ) × D ctr ( C – contra ) −→ k – vect gr is constructed by restrict-ing the functor of tensor product ⊗ B : Hot ( mod – B ) × Hot ( B – mod ) −→ Hot ( k – vect )to the Cartesian product of the homotopy category of CDG-modules that are freeas graded modules and the homotopy category of CDG-modules that are cofree asgraded modules. These definitions of Cotor C , Coext C , and Ctrtor C agree with thedefinitions of the functors Cotor C , Coext C , and Ctrtor C for CDG-coalgebras C byTheorem 6.9.2. Remark.
For any graded vector space C , consider the topological graded tensoralgebra Q n C [ − ⊗ n = lim ←− n L nj =0 C [ − ⊗ j (see Remark 4.4 for the relevant generaldefinitions). One can define a noncounital (“uncurved”) A ∞ -coalgebra C as a struc-ture of augmented DG-algebra with a continuous differential on Q n C [ − ⊗ n . Sucha structure is given by a sequence of linear maps µ n : C −→ C ⊗ n , n = 1, 2, . . .without any convergence condition imposed on them (but satisfying a sequence ofquadratic equations corresponding to the equation d = 0). A morphism of non-counital A ∞ -coalgebras is a continuous morphism of the corresponding topologicalDG-algebras (which always preserves the augmentations). The notion of a non-counital A ∞ -coalgebra is neither more nor less general than that of a noncounitalcurved A ∞ -coalgebra. Still, any noncounital curved A ∞ -coalgebra C with µ = 0can be considered as a noncounital A ∞ -coalgebra. A morphism f of noncounitalcurved A ∞ -coalgebras with µ = 0 can be considered as a morphism of noncouni-tal A ∞ -coalgebras provided that f = 0. A noncounital left A ∞ -comodule M over C is a structure of DG-module with a continuous differential over the topologicalDG-algebra Q n C [ − ⊗ n on the free topological graded module Q n C [ − ⊗ n ⊗ k M .Such a structure is given by a sequence of linear maps λ n : M −→ C ⊗ n ⊗ k M , n = 0, 1, . . . without any convergence conditions imposed. A noncounital leftA ∞ -contramodule P over C is a structure of DG-module over Q n C [ − ⊗ n on thecofree discrete graded module L n Hom k ( C [ − ⊗ n , P ) of continuous homogeneous lin-ear maps Q n C [ − ⊗ n −→ P , where P is discrete. Such a structure is given by asequence of linear maps π n : Hom k ( C ⊗ n , P ) −→ P , n = 0, 1, . . . All the definitionsof 7.4–7.5 are applicable in this situation, and all the results of 7.5 hold in this case.For a (strictly counital or noncounital) curved A ∞ -coalgebra C with µ = 0 there areforgetful functors from the DG-categories of (strictly counital or noncounital) curvedA ∞ -comodules and A ∞ -contramodules to the corresponding DG-categories of un-curved A ∞ -comodules and A ∞ -contramodules. Furthermore, let C be an (uncurved)strictly counital A ∞ -coalgebra. Define the derived categories of strictly counitalA ∞ -comodules and A ∞ -contramodules as the quotient categories of the homotopycategories corresponding to the DG-categories of strictly counital A ∞ -comodules andA ∞ -contramodules by the thick subcategories formed by all the A ∞ -comodules andA ∞ -contramodules that are acyclic with respect to λ and π . Then one can define the unctors Cotor C,I , Coext IC , Ctrtor C,I , Ext IC , and Ext C,I on the Cartesian products ofthe derived categories of strictly counital A ∞ -comodules and A ∞ -contramodules byapplying the functors of topological tensor product and continuous homomorphismsover Q n C + [ − ⊗ n to the corresponding (topological or discrete) CDG-modules over Q n C + [ − ⊗ n . These functors are even preserved by the restrictions of scalars corre-sponding to quasi-isomorphisms of strictly counital A ∞ -coalgebras (i. e., morphisms f such that f is a quasi-isomorphism of complexes with respect to µ ). All theseassertions follow from the result of [14], just as in the proof of Theorem 2.5. In thecase of a strictly counital A ∞ -coalgebra coming from a DG-coalgebra C , the abovefunctors agree with the derived functors Cotor C,I , Coext IC , etc., defined in 2.5.8. Model Categories of DG-Modules,CDG-Comodules, and CDG-Contramodules
By a model category we mean a model category in the sense of Hovey [23].8.1.
Two DG-module model category structures.
Let A = ( A, d ) be a DG-ringand Z DG ( A – mod ) be the abelian category of left DG-modules over A and closedmorphisms between them. Denote by Z DG ( A – mod ) proj and Z DG ( A – mod proj ) thetwo full subcategories of Z DG ( A – mod ) consisting of all the projective DG-modulesin the sense of 1.4 and all the DG-modules that are projective as graded A -modules,respectively. Analogously, denote by Z DG ( A – mod ) inj and Z DG ( A – mod inj ) the twofull subcategories of Z DG ( A – mod ) consisting of all the injective DG-modules in thesense of 1.5 and all the DG-modules that are injective as graded A -modules.It follows from the constructions in the proofs of Theorems 1.4–1.5 that everyobject of Z DG ( A – mod ) proj is homotopy equivalent to an object of Z DG ( A – mod ) proj ∩ Z DG ( A – mod proj ) and every object of Z DG ( A – mod ) inj is homotopy equivalent to anobject of Z DG ( A – mod ) inj ∩ Z DG ( A – mod inj ). Theorem. (a)
There exists a model category structure on the category Z DG ( A – mod ) with the following properties. A morphism is a weak equivalence if and only if it is aquasi-isomorphism. A morphism is a cofibration if and only if it is injective and itscokernel belongs to Z DG ( A – mod ) proj ∩ Z DG ( A – mod proj ) . A morphism is a fibrationif and only if it is surjective. An object is cofibrant if and only if it belongs to Z DG ( A – mod ) proj ∩ Z DG ( A – mod proj ) . All objects are fibrant. (b) There exists a model category structure on the category Z DG ( A – mod ) with thefollowing properties. A morphism is a weak equivalence if and only if it is a quasi-isomorphism. A morphism is a cofibration if and only if it is injective. A morphismis a fibration if and only if it is surjective and its kernel belongs to Z DG ( A – mod ) inj ∩ DG ( A – mod inj ) . All objects are cofibrant. An object is fibrant if and only if it belongsto Z DG ( A – mod ) inj ∩ Z DG ( A – mod inj ) . We will call the model structure of part (a) of Theorem the projective model struc-ture and the model structure of part (b) the injective model structure on the categoryof DG-modules Z DG ( A – mod ). Proof.
We will prove part (a); the proof of (b) is dual. It is clear that all limits andcolimits exist in the abelian category Z DG ( A – mod ). The two-out-of-three axiomfor weak equivalences is obvious, as is the retraction axiom for weak equivalencesand fibrations. The retraction axiom for cofibrations holds since the subcategory Z DG ( A – mod ) proj ∩ Z DG ( A – mod proj ) ⊂ Z DG ( A – mod ) is closed under direct sum-mands. To prove the lifting properties, use [48, Lemma 9.1.1]. One has to checkthat the group Ext ( E, K ) computed in the abelian category Z DG ( A – mod ) vanisheswhenever K ∈ Z DG ( A – mod ) proj ∩ Z DG ( A – mod proj ) and either of the DG-modules E and K is acyclic. Indeed, let E −→ M −→ K be such an extension. Since K ∈ Z DG ( A – mod proj ), the extension of graded A -modules E −→ M −→ K splits.So our extension of DG-modules is given by a closed morphism K −→ E . Since K ∈ Z DG ( A – mod ) proj and one of the DG-modules E and K is acyclic, this closedmorphism is homotopic to zero. It remains to construct the functorial factorizations.For any complex of abelian groups N , let N ′ −→ N be a functorial closed surjec-tive map of complexes of abelian groups such that the induced map of cohomologygroups is surjective and N ′ is a complex of free abelian groups with free cohomologygroups. E. g., one can take N ′ to be the direct sum of Z x over all nonzero homo-geneous elements x of N . Let f : L −→ M be a closed morphism of DG-modulesover A . Then L −→ L ⊕ A ⊗ Z M ′ −→ M is a decomposition of f into a cofibrationfollowed by a fibration and L −→ L ⊕ cone(id A )[ − ⊗ Z M ′ −→ M is a decompositionof f into a trivial cofibration followed by a fibration. To construct a decompositionof f into a cofibration followed by a trivial fibration, start with its decomposition L −→ E −→ M into a cofibration followed by a fibration. Let M be the kernel ofthe closed morphism of DG-modules E −→ M ; set E = A ⊗ Z M ′ . Denote by M the kernel of the closed morphism E −→ E , etc. Let K be the total DG-moduleof the complex of DG-modules · · · −→ E −→ E −→ E formed by taking infinitedirect sums. Then L −→ K is a cofibration and K −→ M is a trivial fibration forthe reasons explained in the proof of Theorem 1.4. (cid:3) CDG-comodule and CDG-contramodule model category structures.
Let C be a CDG-coalgebra over a field k . Let Z DG ( C – comod ) and Z DG ( C – contra )be the abelian categories of left CDG-comodules and left CDG-contramodulesover C with closed morphisms between them. Denote by Z DG ( C – comod inj ) and Z DG ( C – contra proj ) the full subcategories of Z DG ( C – comod ) and Z DG ( C – contra ) ormed by all the CDG-comodules that are injective as graded C -comodules and allthe CDG-contramodules that are projective as graded C -contramodules. Theorem. (a)
There exists a model category structure on the category Z DG ( C – comod ) with the following properties. A morphism is a weak equiva-lence if and only if its cone is a coacyclic CDG-comodule over C . A morphism is acofibration if and only if it is injective. A morphism is a fibration if and only if it issurjective and its kernel belongs to Z DG ( C – comod inj ) . All objects are cofibrant. Anobject is fibrant if and only if it belongs to Z DG ( C – comod inj ) . (b) There exists a model category structure on the category Z DG ( C – contra ) withthe following properties. A morphism is a weak equivalence if and only if its cone isa contraacyclic CDG-contramodule over C . A morphism is a cofibration if and onlyif it is injective and its cokernel belongs to Z DG ( C – contra proj ) . A morphism is afibration if and only if it is surjective. An object is cofibrant if and only if it belongsto Z DG ( C – contra proj ) . All objects are fibrant.Proof. We will prove part (a). All limits and colimits exist in the category Z DG ( C – comod ), since it is an abelian category with infinite direct sums andproducts. The two-out-of-three axiom for weak equivalences holds since coacyclicCDG-comodules form a triangulated subcategory of Hot ( C – comod ). The retractionaxiom for weak equivalences and cofibrations is clear, and for fibrations it followsfrom the fact that the subcategory Z DG ( C – comod inj ) ⊂ Z DG ( C – comod ) is closedunder direct summands. To check the lifting properties, we use the same Lemmathat in the previous proof together with Theorem 4.4(a). Finally, let us constructthe functorial factorizations. To decompose a closed morphism f : L −→ M into acofibration followed by a fibration, find an injective closed morphism from L into ana CDG-comodule J that is injective as a graded C -comodule. This can be done byusing either the comodule version of the functor G − from the proof of Theorem 3.6,or the construction from the proof of Theorem 4.4. Then the morphism f can bepresented as the composition L −→ J ⊕ M −→ M . To obtain a decomposition of f into a cofibration followed by a trivial fibration, one needs to make sure that theCDG-comodule J is contractible. If one uses the functor G − , it suffices to noticethat its image consists entirely of contractible CDG-comodules, and if one uses theconstruction from the proof of Theorem 4.4, one has to replace J with cone(id J ). Toconstruct a decomposition of f into a trivial cofibration followed by a fibration, startwith its decomposition L −→ E −→ M into a cofibration followed by a fibration.Let L − be the cokernel of the morphism L −→ E and L − −→ E − be a closed mor-phism with a coacyclic cone from L − to a CDG-comodule E − ∈ Z DG ( C – comod inj ),which can be constructed in the way of either Theorem 3.7 or Theorem 4.4. Set K = cone( E −→ E − )[ − L −→ K is a trivial cofibration andthe morphism K −→ M is a fibration. (cid:3) emark. Let C be a DG-coalgebra. The model category structure of the firstkind on the abelian category of DG-comodules over C is defined as follows. Let Z DG ( C – comod ) inj ⊂ Z DG ( C – comod ) be the full subcategory consisting of all in-jective DG-comodules over C in the sense of 2.4. Since any coacyclic DG-comoduleis acyclic, every object of Z DG ( C – comod ) inj is homotopy equivalent to an object of Z DG ( C – comod ) inj ∩ Z DG ( C – comod inj ). Let weak equivalences in Z DG ( C – comod )be quasi-isomorphisms, cofibrations be injective morphisms, and fibrations be sur-jective morphisms whose kernels belong to Z DG ( C – comod ) inj ∩ Z DG ( C – comod inj ).One proves that this is a model category structure in the same way as in theabove Theorem, using the facts that Acycl ( C – comod ) and Z DG ( C – comod ) inj ∩ Z DG ( C – comod inj ) form a semiorthogonal decomposition of Hot ( C – comod ), any in-jective closed morphism from an object of Z DG ( C – comod inj ) into an object of Z DG ( C – comod ) splits as an injective morphism of graded comodules over C ,and any object of Z DG ( C – comod ) admits a closed injection into an object of Z DG ( C – comod inj ). The model category structure of the first kind on the abeliancategory of DG-contramodules Z DG ( C – contra ) over C is defined in the dual way.8.3. Finite homological dimension CDG-module case.
Let B be a CDG-ringand Z DG ( B – mod ) be the abelian category of CDG-modules over B and closed mor-phisms between them. Denote by Z DG ( B – mod proj ) and Z DG ( B – mod inj ) the fullsubcategories of Z DG ( B – mod ) formed by all the CDG-modules that are projectiveas graded B -modules and injective as graded B -modules.Assume that the graded ring B has a finite left homological dimension. Theorem. (a)
There exists a model category structure on the category Z DG ( B – mod ) with the following properties. A morphism is a weak equivalence if and only if its coneis absolutely acyclic. A morphism is a cofibration if and only if it is injective and itscokernel belongs to Z DG ( B – mod proj ) . A morphism is a fibration if and only if it issurjective. An object is cofibrant if and only if it belongs to Z DG ( B – mod proj ) . Allobjects are fibrant. (b) There exists a model category structure on the category Z DG ( B – mod ) with thefollowing properties. A morphism is a weak equivalence if and only if its cone is abso-lutely acyclic. A morphism is a cofibration if and only if it is injective. A morphism isa fibration if and only if it is surjective and its kernel belongs to Z DG ( B – mod inj ) . Allobjects are cofibrant. An object is fibrant if and only if it belongs to Z DG ( B – mod inj ) . We will call the model structure of part (a) of Theorem the projective model struc-ture of the second kind and the model structure of part (b) the injective model struc-ture of the second kind on the category of CDG-modules Z DG ( B – mod ). Proof.
Analogous to the proof of Theorem 8.2. Let us spell out some details forpart (a). To check the lifting properties, use the same Lemma from [48] together with heorem 3.5. To construct the functorial factorizations, choose any functor assigningto a graded left B -module N a surjective morphism onto it from a projective gradedleft B -module F ( N ). E. g., one can take F ( N ) to be the direct sum on B x overall homogeneous elements x ∈ N . To decompose a closed morphism f : L −→ M intoa trivial cofibration followed by a fibration, consider the surjective closed morphismonto M from the CDG-module P = G + ( F ( M )). The CDG-module P is projectiveas a graded B -module and contractible, so L −→ L ⊕ P −→ M is the desiredfactorization. To obtain a decomposition of f into a cofibration followed by a trivialfibration, start with a decomposition L −→ E −→ M of f into a cofibration followedby a fibration. Set M = ker( E −→ M ) and choose a closed morphism E −→ M with an absolutely acyclic cone from a CDG-module E ∈ Z DG ( B – mod proj ) into M .Such a morphism exists by Theorem 3.6(b). Set K = cone( E −→ E ); then L −→ K is a cofibration and K −→ M is a trivial fibration. (cid:3) Remark.
For a CDG-ring B such that the graded ring B satisfies the condition ( ∗ )of 3.7, one can define the injective model structure of the second kind on the cat-egory of CDG-modules Z DG ( B – mod ). In this model structure, weak equivalencesare morphisms with coacyclic cones. Cofibrations are injective morphisms, and fi-brations are surjective morphisms whose kernels belong to Z DG ( B – mod inj ). Analo-gously, for a CDG-ring B such that the graded ring B satisfies the condition ( ∗∗ )of 3.8, one can define the projective model structure of the second kind on the cate-gory Z DG ( B – mod ). In this model structure, weak equivalences are morphisms withcontraacyclic cones. Cofibrations are injective morphisms whose cokernels belong to Z DG ( B – mod proj ), and fibrations are surjective morphisms. The proofs are analogousto the above and based on Theorems 3.7–3.8.8.4. Quillen equivalences.
For a definition of Quillen adjunctions and equiva-lences, see [23]. Below we list the important Quillen equivalences and adjunctionsarising from the constructions of Sections 1–6. In order to convey the informationabout the directions of the adjoint pairs, we will always mention the left adjoint func-tor first and the right adjoint functor second. Also we will mention the category thatis the source of the left adjoint functor first, and the category that is the source of theright adjoint functor second. No proofs are given, as they are all very straightforward.For any DG-ring A , the projective and injective model category structures on thecategory of DG-modules Z DG ( A – mod ) are Quillen equivalent. The equivalence isprovided by the adjoint pair of identity functors between Z DG ( A – mod ) and itself,where the identity functor from Z DG ( A – mod ) with its projective model structureto Z DG ( A – mod ) with its injective model structure is considered as the left adjointfunctor, and the identity functor in the other direction is considered as the rightadjoint functor. nalogously, for any CDG-ring B such that the graded ring B has finite lefthomological dimension, the projective and injective model category structures onthe category of CDG-modules Z DG ( B – mod ) are Quillen equivalent. The equiva-lence is provided by the adjoint pair of identity functors, where the identity func-tor from Z DG ( B – mod ) with its projective model structure of the second kind to Z DG ( B – mod ) with its injective model structure of the second kind is considered asthe left adjoint functor, and the identity functor in the other direction is consideredas the right adjoint functor.The comodule-contramodule correspondence for a CDG-coalgebra C over a field k can also be understood as a Quillen equivalence. The pair of adjoint func-tors Φ C : Z DG ( C – contra ) −→ Z DG ( C – comod ) and Ψ C : Z DG ( C – contra ) −→ Z DG ( C – comod ) is a Quillen equivalence between the model category of leftCDG-contramodules over C and the model category of left CDG-comodules over C .Let τ : C −→ A be an acyclic twisting cochain between a conilpotent CDG-coal-gebra C and a DG-algebra A . Then the adjoint pair of Koszul duality functors N A ⊗ τ N and M C ⊗ τ M is a Quillen equivalence between the modelcategory Z DG ( C – comod ) of left CDG-comodules over C and the model category Z DG ( A – mod ) of left DG-modules over A , with the projective model structureon the latter. The adjoint pair of Koszul duality functors P Hom τ ( C, P )and Q Hom τ ( A, Q ) is a Quillen equivalnce between the model category Z DG ( A – mod ) of left DG-modules over A and the model category Z DG ( C – contra )of left CDG-contramodules over C , with the injective model structure on the former.Analogously, let C be a CDG-coalgebra and τ = τ C,w : C −→ Cob w ( C ) bethe natural twisting cochain. Then the adjoint pair of Koszul duality functors N B ⊗ τ N and M C ⊗ τ M is a Quillen equivalence between the modelcategory Z DG ( C – comod ) of left CDG-comodules over C and the model category Z DG ( A – mod ) of left CDG-modules over B , with the projective model structureon the latter. The adjoint pair of Koszul duality functors P Hom τ ( C, P )and Q Hom τ ( B, Q ) is a Quillen equivalnce between the model category Z DG ( B – mod ) of left DG-modules over B and the model category Z DG ( C – contra )of left CDG-contramodules over C , with the injective model structure on the former.Notice that in each of the two Koszul duality situations above, the conilpotentand the nonconilpotent one, we have four model categories, two of them on theDG-algebra or CDG-algebra side and two on the CDG-coalgebra side. The fourQuillen equivalences between these four model categories form a circular diagram:if one looks at the direction of, e. g., the left adjoint functor in each of the adjointpairs, one finds that these functors map Z DG ( A – mod ) or Z DG ( B – mod ) with theprojective model structure to the same category with the injective model structure to Z DG ( C – contra ) to Z DG ( C – comod ) and back to Z DG ( A – mod ) or Z DG ( B – mod )with the projective model structure. et f : A −→ B be a morphism of DG-algebras (see 1.7). Then the pair of ad-joint functors E f : Z DG ( A – mod ) −→ Z DG ( B – mod ) and R f : Z DG ( B – mod ) −→ Z DG ( A – mod ) is a Quillen adjunction between the model categories Z DG ( A – mod )and Z DG ( B – mod ) with the projective model structures. The pair of adjointfunctors R f : Z DG ( B – mod ) −→ Z DG ( A – mod ) and E f : Z DG ( A – mod ) −→ Z DG ( B – mod ) is a Quillen adjunction between the model categories Z DG ( B – mod )and Z DG ( A – mod ) with the injective model structures.Let f : C −→ D be a morphism of CDG-coalgebras (see 4.8). Then the pairof adjoint functors R f : Z DG ( C – comod ) −→ Z DG ( D – comod ) and E f : Z DG ( D – comod ) −→ Z DG ( C – comod ) is a Quillen adjunction between the modelcategories of CDG-comodules over C and CDG-comodules over D . The pairof adjoint functors E f : Z DG ( D – contra ) −→ Z DG ( C – contra ) and R f : Z DG ( C – contra ) −→ Z DG ( D – contra ) is a Quillen adjunction between the model cate-gories of CDG-contramodules over D and CDG-contramodules over C .9. Model Categories of DG-Algebras and CDG-Coalgebras
Model category of DG-algebras.
Let k be a commutative ring. Denote by k – alg dg the category of DG-algebras over k and by k – alg augdg the category of augmentedDG-algebras over k , i. e., DG-algebras endowed with a DG-algebra morphism onto k .Given a DG-algebra A , we will consider DG-algebras of the special form A h x n,α i .These are DG-algebras that, as graded algebras, are obtained by adjoining to A adouble-indexes family of free homogeneous generators x n,α , where n = 1, 2, . . . and α belongs to some index set. The differential on A h x n,α i must satisfy the conditionsthat A is a DG-subalgebra of A h x n,α i and for any adjoined free generator x n,α theelement d ( x n,α ) belongs to the subalgebra generated by A and x m,β with m < n .We will also consider DG-algebras of the even more special form A h x ω , dx ω i . Theseare DG-algebras that, as graded algebras, are obtained by adjoining to A the freehomogeneous generators x ω and dx ω ; the differential on A h x ω , dx ω i is defined in theway that the notation suggests. So A h x ω , dx ω i can be also defined as the DG-algebrafreely generated by a DG-algebra A and the elements x ω .The same notation applies to augmented DG-algebras, with the following obviouschanges. When working with augmented DG-algebras, one adjoins free homogeneousgenerators x n,α or x ω annihilated by the augmentation morphism, and requires thattheir differentials be also annihilated by the augmentation morphism.The following result is due to Hinich and Jardine [21, 26]. Theorem. (a)
There exists a model category structure on k – alg dg with the followingproperties. A morphism is a weak equivalence if and only if it is a quasi-isomorphism.A morphism is a fibration if and only if it is surjective. A morphism is a cofibration f and only if it is a retract of a morphism of the form A −→ A h x n,α i . A morphismis a trivial cofibration if and only if it is a retract of a morphism of the form A −→ A h x ω , dx ω i . All DG-algebras are fibrant. A DG-algebra is cofibrant if and only if itis a retract of a DG-algebra of the form k h x n,α i . (b) The same assertion applies to the category of augmented DG-algebras k – alg augdg .Proof. It is easy to see that all limits and colimits exist in k – alg dg . All of them arepreserved by the forgetful functor to the category of graded algebras; all limits andfiltered colimits are even preserved by the forgetful functor to the category of com-plexes of vector spaces. The two-out-of-three axiom for weak equivalences and theretraction axiom for all three classes of morphisms are obvious. It is straightforwardto check that any morphism of the form A −→ A h x ω , dx ω i is a quasi-isomorphism. Todecompose a morphism of DG-algebras A −→ B as A −→ A h x ω , dx ω i −→ B with asurjective morphism A h x ω , dx ω i −→ B , it suffices to have generators x ω correspond-ing to all the homogeneous elements of B . To decompose a morphism A −→ B as A −→ A h x n,α i −→ B with a surjective quasi-isomorphism A h x n,α i −→ B , the follow-ing inductive process is used. Let x ,α correspond to all the homogeneous elements of B annihilated by the differential, and x ,β correspond to all the homogeneous elementsof B . Then the morphism A h x ,α , x ,β i −→ B is surjective and the induced morphismon the cohomology is also surjective. Furthermore, let x ,γ correspond to all homoge-neous cocycles in A h x ,α , x ,β i whose images are coboundaries in B , let x ,δ correspondto all homogeneous cocycles in A h x ,α , x ,β , x ,γ i whose images are coboundaries in B , etc. The lifting property of the morphisms A −→ A h x ω , dx ω i with respect tosurjective morphisms of DG-algebras is obvious, and the lifting property of the mor-phisms A −→ A h x n,α i with respect to surjective quasi-isomorphisms is verified byinduction on n and a straightforward diagram chase. It remains to check that all thequasi-isomorphisms of DG-algebras that are retracts of morphisms E −→ E h x n,α i areactually retracts of morphisms A −→ A h x ω , dx ω i . Let f : A −→ B be such a quasi-isomorphism; decompose it as A −→ A h x ω , dx ω i −→ B , where A h x ω , dx ω i −→ B is a surjective morphism. Then the latter morphism is also a quasi-isomorphism bythe two-out-of-three property of quasi-isomorphisms. Therefore, the morphism f ,being a retract of a morphism E −→ E h x n,α i , has the lifting property with respectto the morphism A h x ω , dx ω i −→ B . It follows that f is a retract of the morphism A −→ A h x ω , dx ω i . (cid:3) Let k be a field. For the purposes of nonaugmented Koszul duality one needsto consider the full subcategory k – alg + dg of k – alg dg formed by all the DG-algebraswith nonzero units. This full subcategory is obtained from the category k – alg dg byexcluding its final object A = 0. There are no morphisms from this final object toany objects in k – alg dg that are not final. The category k – alg + dg is a model categorywithout limits, i. e., the classes of morphisms in k – alg + dg that are weak equivalences, ofibrations, and fibrations in k – alg dg satisfy all the axioms of model category exceptfor the existence of limits and colimits. In fact, limits of all nonempty diagrams, allcoproducts, and all filtered colimits exist in k – alg + dg . At the same time, k – alg + dg hasno final object, and fibered coproducts sometimes do not exist in it.9.2. Limits and colimits of CDG-coalgebras.
Let k be a field. We will use thenotation and terminology of 6.10. The forgetful functor from the category of conilpo-tent or coaugmented coalgebras to the category of graded vector spaces assigns toa coaugmented DG-coalgebra ( C, w ) the graded vector space
C/w ( k ). The forget-ful functor from the category of conilpotent or coaugmented DG-coalgebras to thecategory of complexes of vector spaces is defined in the analogous way. Lemma. (a)
Limits of all nonempty diagrams, all coproducts, and all filtered col-imits exist in the category of conilpotent CDG-coalgebras k – coalg conilpcdg . All limits ofdiagrams with a final vertex, and consequently all filtered limits, are preserved by theforgetful functor to the category of conilpotent graded coalgebras. All coproducts andfiltered colimits are preserved by the forgetful functors to the categories of conilpotentgraded coalgebras, coaugmented graded coalgebras, and graded vector spaces. (b) All limits and colimits exist in the category of conilpotent DG-coalgebras k – coalg conilpdg . All of them are preserved by the forgetful functor to the category ofconilpotent graded coalgebras. All colimits are preserved by the forgetful functor tothe categories of coagmented graded coalgebras and complexes of vector spaces.Proof. Let us start with considering limits and colimits in the category of coaug-mented graded coalgebras k – coalg coaug . It is easy to see that colimits exist in k – coalg coaug and commute with the forgetful functor to the category of graded vec-tor spaces. To construct limits in k – coalg coaug , it suffices to obtain finite products,filtered limits, and equalizers. Finite products in k – coalg coaug are coaugmented coal-gebras cofreely cogenerated by the coaugmented coalgebras being multiplied [1]. Fil-tered limits and equalizers in k – coalg coaug are graded vector subspaces of the filteredlimits and equalizers in the category of graded vector spaces. In particular, thefiltered limit of a diagram of coalgebras C α is the subspace of its limit as a dia-gram of graded vector spaces lim ←− C α equal to the full preimage of the tensor prod-uct lim ←− C α ⊗ k lim ←− C α ⊂ lim ←− C α ⊗ k C α under the limit of the comultiplication mapslim ←− C α −→ lim ←− C α ⊗ k C α . The same applies to limits and colimits in the category ofgraded coalgebras k – coalg (with its different forgetful functor). Colimits in the cate-gory of conilpotent coalgebras k – coalg conilp are compatible with those in k – coalg coaug ,and limits in k – coalg conilp are the maximal conilpotent subcoalgebras of the limits in k – coalg coaug . A simpler approach is to construct directly finite products in k – coalg conilp as the cofreely cogenerated conilpotent coalgebras. his suffices to clarify part (b), so let us turn to (a). Any diagram with a final vertexin k – coalg conilpcdg is isomorphic to a diagram of strict morphisms, i. e., morphisms of theform ( f, k – coalg conilpcdg agrees with that in k – coalg conilp . Products ofpairs of objects and equalizers in k – coalg conilpcdg have to be constructed explicitly. Here itis instructive to start with the dual case of coproducts of pairs and coequalizers in thecategory of CDG-algebras k – alg cdg . Given two morphisms ( f ′ , a ′ ) and ( f ′′ , a ′′ ) : B −→ A in k – alg cdg , one constructs their coequalizer as the quotient algebra of A by thetwo-sided ideal generated by all the elements f ′ ( b ) − f ′′ ( b ) and a ′ − a ′′ ∈ A , where b ∈ B . This is a certain graded quotient algebra of the coequalizer of f ′ and f ′′ in the category of graded algebras k – alg . Given two CDG-algebras A and B , oneconstructs their coproduct A ⊔ B as the graded algebra freely generated by A , B , andan element c of degree 1. The restriction of the differential in A ⊔ B to A coincideswith d A and its restriction to B is given by the formula b d B ( b ) − [ c, b ] for b ∈ B .The action of the differential in A ⊔ B on the element c is obtained from the equation h A + dc + c = h B . The constructions in k – coalg conilpcdg are dual.Any diagram of a filtered inductive limit in k – coalg conilpcdg is isomorphic to a diagramof strict morphisms. This follows from the vanishing of the first derived functorof filtered colimit in k – vect computed in terms of the standard bar-construction.Colimits of all diagrams of strict morphisms exist in k – coalg conilpcdg and are preservedby the three forgetful functors. (cid:3) The category k – coalg conilpcdg has no final object, and fibered coproducts sometimesdo not exist in it. The closest candidate for a final object in k – coalg conilpcdg is thefollowing CDG-coalgebra ( D, d, h ). The coalgebra D is cofreely cogenerated by thelinear function h , i. e., it is the graded tensor coalgebra of a one-dimensional vectorspace concentrated in degree −
2. The differential d is zero. For any CDG-coalgebra D there is a morphism of CDG-coalgebras C −→ D ; however, such a morphism is notunique. The CDG-coalgebra D is Koszul dual to a DG-algebra with zero cohomology.Notice also that ( D, d, h ) is the final object in the category of conilpotentCDG-coalgebras and strict morphisms between them. It follows from the aboveproof that all limits and colimits exist in the latter category.9.3.
Model category of CDG-coalgebras.
In this subsection we presume all coal-gebras to be conilpotent. In particular, by the graded coalgebra cofreely cogeneratedby a graded vector space, or the graded coalgebra cofreely cogenerated by anothergraded coalgebra and a graded vector space, we mean the corresponding universalobjects in the category of conilpotent graded coalgebras. These in general differ fromthe analogous universal objects in the categories of arbitrary (or coaugmented) gradedcoalgebras. The same applies to the DG-coalgebra cofreely cogenerated by another
G-coalgebra and a complex of vector spaces. Notice that the latter constructiontransforms quasi-isomorphisms in either of its arguments to quasi-isomorphisms.All increasing filtrations below are presumed to be cocomplete.For any category C , denote by C fin the category obtained by formal adjoining of afinal object ∗ to C . By the definition, there is one morphism into ∗ from any objectof C fin , and there are no morphisms from ∗ into any object of C but ∗ itself. We referto the category C fin as the finalized category C . Theorem. (a)
There is a model category structure on the finalized category of conilpo-tent CDG-coalgebras k – coalg conilpcdg , fin with the following properties. A morphism ofCDG-coalgebras is a weak equivalence if and only if it belongs to the minimal classof morphisms containing the filtered quasi-isomorphisms and satisfying the two-out-of-three axiom. Also a morphism of CDG-coalgebras is a weak equivalence if andonly if it is a composition of retracts of filtered quasi-isomorphisms. A morphismof CDG-coalgebras is a cofibration if and only if the underlying morphism of gradedcoalgebras is injective. A morphism of CDG-coalgebras C −→ D is a fibration if andonly if the graded coalgebra C is cofreely cogenerated by the graded coalgebra D and a graded vector space. A morphism of CDG-coalgebras is a trivial cofibration ifand only if it belongs to the minimal class of morphisms which contains the injectivefiltered quasi-isomorphisms strictly compatible with the filtrations, is closed under thecomposition, and contains a morphism g whenever it contains morphisms f and f g .Also a morphism of CDG-coalgebras is a trivial cofibration if and only if it is a retractof an injective filtered quasi-isomorphism strictly compatible with the filtrations. Amorphism of CDG-coalgebras is a trivial fibration if and only if it is a retract of a mor-phism of CDG-coalgebras C −→ D such that C and D admit increasing filtrations F compatible with the comultiplications and differentials such that F C = w ( k ) = F D and the DG-coalgebra gr F C is cofreely cogenerated by the DG-coalgebra gr F D and anacyclic complex of vector spaces. The only morphism from any CDG-coalgebra to theobject ∗ is a cofibration, but not a weak equivalence. All objects of k – coalg conilpcdg , fin arecofibrant. A CDG-coalgebra is fibrant if and only if its underlying graded coalgebra isa cofree conilpotent coalgebra (graded tensor coalgebra). (b) There is a model category structure of the category of conilpotent DG-coalgebras k – coalg conilpdg with the following properties. A morphism of DG-coalgebras is a weakequivalence if and only if it belongs to the minimal class of morphisms containingthe filtered quasi-isomorphisms and satisfying the two-out-of-three axiom. Also amorphism of DG-coalgebras is a weak equivalence if and only if it is a composition ofretracts of filtered quasi-isomorphisms. A morphism of DG-coalgebras is a cofibrationif and only if it is injective. A morphism of DG-coalgebras C −→ D is a fibrationif and only if the graded coalgebra C is cofreely cogenerated by the graded coalgebra D and a graded vector space. A morphism of DG-coalgebras is a trivial cofibration if nd only if it belongs to the minimal class of morphisms which contains the injectivefiltered quasi-isomorphisms strictly compatible with the filtrations, is closed under thecomposition, and contains a morphism g whenever it contains morphisms f and f g .Also a morphism of DG-coalgebras is a trivial cofibration if and only if it is a retractof an injective filtered quasi-isomorphism strictly compatible with the filtrations. Amorphism of DG-coalgebras is a trivial fibration if and only if it is a retract of amorphism of DG-coalgebras C −→ D such that C and D admit increasing filtrations F compatible with the comultiplications and differentials such that F C = w ( k ) = F D and the DG-coalgebra gr F C is cofreely cogenerated by the DG-coalgebra gr F D and an acyclic complex of vector spaces. All objects of k – coalg conilpdg are cofibrant.A DG-coalgebra is fibrant if and only if its underlying graded coalgebra is a cofreeconilpotent coalgebra (graded tensor coalgebra).Proof. We will prove part (a). Existence of limits and colimits in k – coalg conilpcdg , fin followsfrom Lemma 9.2. It suffices to check that coequalizers exist in k – coalg conilpcdg , fin .In the rest of the proof we use the Koszul duality functors Bar v and Cob w to-gether with Theorem 9.1. It will follow from our argument that the weak equiva-lences, fibrations, and trivial fibrations of CDG-coalgebras can be also characterizedin the following ways. A morphism of CDG-coalgebras is a weak equivalence if andonly if the functor Cob w transforms it into a quasi-isomorphism. A morphism ofCDG-coalgebras is a fibration if and only if it is a retract of a morphism obtained bya base change from a morphism obtained by applying the functor Bar v to a fibrationof DG-algebras. A morphism of CDG-coalgebras is a trivial fibration if and only ifit is a retract of a morphism obtained by a base change from a morphism obtainedby applying the functor Bar v to a trivial fibration of DG-algebras.The proof is based on several Lemmas. Lemma 1. (i)
Any morphism of CDG-coalgebras C −→ D can be decomposed intoan injective morphism of CDG-coalgebras C −→ E followed by a morphism E −→ D obtained by a base change from a morphism obtained by applying the functor Bar v toa surjective quasi-isomorphism of DG-algebras. Furthermore, the CDG-coalgebras E and D admit increasing filtrations F compatible with the comultiplications and differ-entials such that F E = k = F D and the DG-coalgebra gr F E is cofreely cogeneratedby the DG-coalgebra gr F D and an acyclic complex of vector spaces. (ii) Any morphism of CDG-coalgebras C −→ D can be can be decomposed into aninjective filtered quasi-isomorphism C −→ E strictly compatible with the filtrationsand a morphism E −→ D obtained by a base change from a morphism obtained byapplying the functor Bar v to a surjective morphism of DG-algebras.Proof. This Lemma is based on a construction of Hinich [22]. To prove (i), decom-pose the morphism of DG-algebras Cob w ( C ) −→ Cob w ( D ) into an injective mor-phism Cob w ( C ) −→ A followed by a surjective quasi-isomorphism A −→ Cob w ( D ). onsider the induced morphism Bar v ( A ) −→ Bar v Cob w ( D ) and set E to be thefibered product of the CDG-coalgebras Bar v ( A ) and D over Bar v Cob w ( D ). Then C −→ E −→ D is the desired decomposition. Now let E −→ D be the morphismof CDG-coalgebras obtained from the morphism Bar v ( A ) −→ Bar v ( B ) induced by amorphism of DG-algebras A −→ B by the base change with respect to a morphism ofCDG-coalgebras D −→ Bar v ( B ). Define an increasing filtration F on the DG-algebra A by the rules F A = k , F A = k ⊕ ker( A → B ), and F A = A . Denote also by F the induced filtrations on the quotient DG-algebra B of A and the bar-constructionsBar v ( A ) and Bar v ( B ). Define an increasing filtration F on the CDG-coalgebra D bythe rules F n D = F n +1 D = G n D , where G is the natural increasing filtration definedin 6.4 (where it is denoted by F ). Denote by F the induced filtration on the fiberedproduct E . Then the DG-coalgebra gr F E is the fibered product of the DG-coalgebrasBar v gr F A and gr F D over Bar v gr F B , and the DG-coalgebra Bar v gr F A is cofreely co-generated by Bar v gr F B and an acyclic complex of vector spaces, hence the morphismof DG-algebras gr F E −→ gr F D has the same property.Let us prove part (ii). Let A be the DG-algebra freely generated by the DG-algebraCob w ( C ) and the acyclic complex of vector spaces id Cob w ( D ) [ − w ( C ) −→ Cob w ( D ) factorizes into a trivial cofibrationCob w ( C ) −→ A followed by a fibration A −→ Cob w ( C ). Let F denote the naturalincreasing filtrations on the conilpotent CDG-coalgebras C and D and the induced fil-trations on Cob w ( C ), Cob w ( D ), and A . Then the morphism Cob w (gr F C ) −→ gr F A is an injective quasi-isomorphism and the morphism gr F A −→ Cob w (gr F D ) is sur-jective. Set E to be the fibered product of the CDG-coalgebras Bar v ( A ) and D overBar v Cob w ( D ). Let us show that the decomposition C −→ E −→ D has the de-sired properties. Denote by F the induced filtration on E as a CDG-subcoalgebra ofBar v ( A ). The morphism gr F C −→ Bar v (gr F A ) is an injective quasi-isomorphism,and it only remains to check that the morphism gr F E −→ Bar v (gr F A ) is aquasi-isomorphism. Notice that the DG-coalgebra gr F E is the fibered product ofBar v (gr F A ) and gr F D over Bar v Cob w (gr F D ).Let n denote the nonnegative grading induced from the indexing of the filtra-tion F . Introduce a decreasing filtration G on gr F A by the rules that G gr F A = gr F A and G j gr F A is the sum of the components of the ideal ker(gr F A → Cob w (gr F D ))situated in the grading n > j . This filtration is locally finite with respect tothe grading n . Let G denote the induced decreasing filtration on gr F E as aDG-subcoalgebra of Bar v (gr F A ). Then the DG-coalgebra gr G gr F E is the fiberedproduct of Bar v (gr G gr F A ) and gr F D over Bar v Cob w (gr F D ). The DG-algebragr G gr F A and the DG-coalgebra gr G gr F D have two nonnegative gradings n and j .Introduce an increasing filtration H on the DG-algebra gr G gr F A for which thecomponent H t is the sum of all the components (gr G gr F A ) n,j with j = 0 and > n or j > t >
1. Once again, denote by H the induced increas-ing filtration on gr G gr F E as a DG-subcoalgebra of Bar v (gr G gr F A ). Then theDG-coalgebra gr H gr G gr F E is the fibered product of Bar v (gr H gr G gr F A ) and gr F D over Bar v Cob w (gr F D ). The DG-coalgebra Bar v (gr H gr G gr F A ) is cofreely cogener-ated by Bar v Cob w (gr F D ) and a complex of vector spaces, and the DG-coalgebragr H gr G gr F E is cofreely cogenerated by gr F D and the same complex of vector spaces.Since the morphism gr F D −→ Bar v Cob w (gr F D ) is a quasi-isomorphism, so is themorphism gr H gr G gr F E −→ Bar v (gr H gr G gr F A ). (cid:3) Lemma 2. (i)
Injective filtered quasi-isomorphisms of CDG-coalgebras strictly com-patible with the filtrations have the lifting property with respect to the morphisms ofCDG-coalgebras C −→ D such that the graded coalgebra C is cofreely cogeneratedby the graded coalgebra D and a graded vector space. (ii) The class of morphisms of conilpotent graded coalgebras C −→ D such that C is cofreely generated by D and a graded vector space is closed under retracts. If theassociated quotient morphism of a morphism f with respect to an increasing filtrationbelongs to this class, then the morphism f itself does.Proof. Part (i): let g : X −→ Y be an injective morphism of CDG-coalgebras en-dowed with increasing filtrations F making f a filtered quasi-isomorphism strictlycompatible with the filtrations, i. e., F n X = g − ( F n Y ). Clearly, we can restrictourselves to diagrams of strict morphisms of CDG-coalgebras, i. e., morphisms ofthe form ( f, Y −→ D has been lifted to morphisms X −→ C and F n − Y −→ C in compatible ways; we need to extend the second mor-phism to F n Y in a compatible way. The quotient space F n Y / ( F n − Y + g ( F n X )) is anacyclic complex of vector spaces. Choose a graded subspace V ′ in this complex suchthat the restriction of the differential to V ′ is injective and the complex coincideswith V ′ + d ( V ′ ). Let V be any graded subspace in F n Y which projects isomorphi-cally onto V ′ . By the assumption about the morphism C −→ D , one can extendthe morphism F n − Y −→ C to a graded coalgebra morphism F n − Y + V −→ C in a way compatible with the morphisms into D . Then the condition of compat-ibility with the differentials allows to extend this morphism in the unique way toa strict CDG-coalgebra morphism F n − Y + V + dV −→ C . Finally, one combinesthis morphism with the morphism F n X −→ C to obtain the desired CDG-coalgebramorphism F n Y −→ C .Part (ii): This class of morphisms of conilpotent graded coalgebras is characterizedby the conditions that the induced morphism Ext C ( k, k ) −→ Ext D ( k, k ) is surjectiveand the morphism Ext C ( k, k ) −→ Ext D ( k, k ) is an isomorphism. Notice also thenatural isomorphism Ext iE ( k, k ) ≃ Cotor E − i ( k, k ) for a coaugmented coalgebra E . (cid:3) Lemma 3. (i)
The functor
Cob w maps injective morphisms of CDG-coalgebras tocofibrations of DG-algebras. ii) Injective morphisms of CDG-coalgebras have the lifting property with respectto morphisms obtained by a base change from morphisms obtained by applying thefunctor
Bar v to surjective quasi-isomorphisms of DG-algebras. (iii) Injective morphisms of CDG-coalgebras that are transformed into quasi-isomorphisms by the functor
Cob w have the lifting property with respect to morphismsobtained by a base change from morphisms obtained by applying the functor Bar v tosurjective morphisms of DG-algebras.Proof. Parts (ii) and (iii) follow immediately from part (i) and the adjunction of func-tors Bar v and Cob w . To check (i), notice that for an injective morphism of conilpotentCDG-coalgebras X −→ Y , the DG-algebra Bar v ( Y ) has the form Bar v ( X ) h x n,α i . In-deed, let F denote the natural increasing filtration on Y . Then the DG-algebraBar v ( X + F n Y ) has the form Bar v ( X + F n − Y ) h x ,α , x ,β i . (cid:3) Now we can finish the proof of Theorem.It follows from (the proof of) Theorem 6.10 that the minimal class of morphisms ofCDG-algebras containing the filtered quasi-isomorphisms and satisfying the two-out-of-three axiom coincides with the class of morphisms that are transformed into quasi-isomorphisms by the functor Cob w . Analogously, the minimal class of morphismsthat contains the injective filtered quasi-isomorphisms strictly compatible with thefiltrations, is closed under the composition, and contains a morphism g whenever itcontains morphisms f and f g coincides with the class of injective morphisms thatare transformed into quasi-isomorphisms by the functor Cob w .Let X −→ Y be a morphism of the latter class. Using Lemma 1(ii), decom-pose it into an injective filtered quasi-isomorphisms X −→ E strictly compatiblewith the filtrations, followed by a morphism E −→ Y obtained by a base changefrom a morphism obtained by applying the functor Bar v to a surjective morphism ofDG-algebras. By Lemma 3(iii), the morphism X −→ Y has the lifting property withrespect to the morphism E −→ Y , hence the morphism X −→ Y is a retract of themorphism X −→ E .Let C −→ D be a morphism which is transformed into a quasi-isomorphism by thefunctor Cob w and such that C is cofreely cogenerated by D and a graded vectorspace. Using Lemma 1(i), decompose it into an injective morphism C −→ E followedby a morphism E −→ D obtained by a base change from a morphism obtained byapplying the functor Bar v to a surjective quasi-isomorphism of DG-algebras. Thelatter morphism is also a filtered quasi-isomorphism, and even admits increasingfiltrations F on E and D such that the DG-coalgebra gr F E is cofreely cogeneratedby gr F D and an acyclic complex of vector spaces. Hence the injective morphism C −→ E is also transformed into a quasi-isomorphism by the functor Cob w , andtherefore is a retract of an injective filtered quasi-isomorphism strictly compatiblewith the filtrations. By Lemma 2(i), the morphism C −→ E has the lifting property ith respect to the morphism C −→ D , so the morphism C −→ D is a retract of themorphism E −→ D .Let C −→ D be a morphism that is transformed into a quasi-isomorphism bythe functor Cob w . Using Lemma 1(i) or (ii), one can decompose it into an injectivemorphism C −→ E and a morphism E −→ D such that E is cofreely cogeneratedby D and a graded vector space, and both morphisms C −→ E and E −→ D are transformed into quasi-isomorphisms by the functor Cob w . Thus the morphism C −→ D is the composition of two retracts of filtered quasi-isomorphisms.The rest of the proof is straightforward. In addition to the above Lemmas, one hasto use the analogue of Lemma 2 designed to imply the lifting and retraction propertiesof the morphisms C −→ ∗ , where C is a cofree conilpotent graded coalgebra. (cid:3) Consider the finalized category of DG-algebras with nonzero units k – alg + dg , fin . Ithas a model category structure in which a morphism of DG-algebras is a weak equiv-alence, cofibration, or fibration if and only if it belongs to the corresponding class ofmorphisms in k – alg dg , and for any DG-algebra A the morphism A −→ ∗ is a fibrationand a cofibration, but not a weak equivalence. It follows from the above that thefunctors Cob w and Bar v define a Quillen equivalence between the model categories k – coalg conilpcdg , fin and k – alg + dg , fin . Let us point out that the natural Quillen adjunctionbetween the categories k – alg dg and k – alg + dg , fin is not a Quillen equivalence. Analo-gously, the functors Cob w and Bar v define a Quillen equivalence between the modelcategories k – coalg conilpdg and k – alg augdg .9.4. Cofibrant DG-algebras.
The following result demonstrates the importance ofDG-algebras that are cofibrant in the model category structure of Theorem 9.1(a).Let k be a commutative ring of finite homological dimension. Theorem.
Let A be a cofibrant DG-algebra over k . Then all the four trian-gulated subcategories Acycl abs ( A – mod ) , Acycl co ( A – mod ) , Acycl ctr ( A – mod ) , and Acycl ( A – mod ) ⊂ Hot ( A – mod ) coincide. Consequently, the derived categories of thefirst and the second kind D ( A – mod ) and D abs ( A – mod ) are isomorphic.Proof. The graded ring A has the homological dimension exceeding the homologicaldimension of k by at most 1, being a retract of a free graded algebra over k . Hence Acycl co ( A – mod ) = Acycl abs ( A – mod ) = Acycl ctr ( A – mod ) by Theorem 3.6(a). Let usprove that Acycl abs ( A ) = Acycl ( A ) for a DG-algebra A of the form A = k h x n,α i ; thisis clearly sufficient. Let C + [ −
1] denote the free graded k -module spanned by x n,α ;then C = k ⊕ C + can be considered as a (strictly coaugmented strictly counital)curved A ∞ -coalgebra over k (see 7.5). Set F n C to be the linear span of k and x m,β for m n . Let M be a left DG-module over A . Consider the surjective closedmorphism of DG-modules A ⊗ k M −→ M . Its kernel can be naturally identified with A ⊗ k C + [ − ⊗ k M , so there is the induced DG-module structure on this triple tensor roduct. The cone of the morphism of DG-modules A ⊗ k C + [ − ⊗ k M −→ A ⊗ k M can be naturally identified with A ⊗ k C ⊗ k M , so on the latter triple tensor productthere is also a natural DG-module structure. Consider the filtration F on A ⊗ k C ⊗ k M induced by the filtration F on C . This filtration is compatible with the differentialand the quotient DG-modules ( A ⊗ k F n C ⊗ k M ) / ( A ⊗ k F n − C ⊗ k M ) are isomorphicto the DG-modules A ⊗ k F n C/F n − C ⊗ k M with the differentials induced by thedifferentials on A and M . It follows that the DG-module A ⊗ k C ⊗ k M is projectivein the sense of 1.4 whenever M is a projective complex of k -modules. Furthermore,this DG-module is coacyclic whenever M is coacyclic as a complex of k -modules.Since the cone of the morphism A ⊗ k C ⊗ k M −→ M is always absolutely acyclic andthe ring k has a finite homological dimension, it follows that M is absolutely acyclicwhenever it is acyclic. (cid:3) Let A be a DG-ring for which the underlying graded ring A has a finite lefthomological dimension. When do the triangulated subcategories Acycl ( A ) and Acycl abs ( A ) coincide? This cannot happen too often, as the absolute derived categories D abs ( A – mod ) and D abs ( B – mod ) are isomorphic for any two DG-rings A and B thatare isomorphic as CDG-rings. At the same time, the derived category D ( A – mod )only depends on the quasi-isomorphism class of A . These are two very differentand hardly ever compatible kinds of functoriality. The quasi-isomorphism classesof CDG-isomorphic DG-rings A and B can be entirely unrelated to each other; seeExamples below.Nevertheless, there are the following cases, in addition to the above Theorem andCorollary 6.8.2. When either A i = 0 for all i >
0, or A i = 0 for all i <
0, thering A is semisimple, and A = 0, one has Acycl ( A ) = Acycl abs ( A ). This followsfrom Theorems 3.4.1(d)– 3.4.2(d) and 3.6(a). Besides, using Theorem 3.6(b) one cancheck [32] that Acycl ( A ) = Acycl abs ( A ) whenever A has a zero differential. All of thisassumes that A has a finite left homological dimension; without this assumption,there are only some partial results obtained in 3.4. Examples.
Let V be a (totally) finite-dimensional complex of k -vector spaces and A be the graded algebra of endomorphisms of V endowed with the induced differential d .Then the DG-algebra ( A, d ) is isomorphic to the DG-algebra ( A,
0) in the categoryof CDG-algebras over k . In particular, when V is acyclic but nonzero, ( A, d ) is alsoacyclic, while ( A,
0) is not. So the derived category D (( A, d )– mod ) vanishes, while thederived category D (( A, mod ) is equivalent to D ( k – vect ). At the same time, theabsolute derived categories D abs (( A, d )– mod ) and D abs (( A, mod ) are isomorphic.Furthermore, let ( D ′ , d ′ , h ′ ) and ( D ′′ , d ′′ , h ′′ ) be two coaugmented CDG-coalgebraswith coaugmentations w ; then the CDG-coalgebra ( D ′ ⊕ D ′′ , d ′ + d ′′ , h ′ + h ′′ ) hastwo induced coaugmentations w ′ and w ′′ . The DG-algebra Cob w ′ ( D ′ ⊕ D ′′ ) is quasi-isomorphic to Cob w ( D ′ ) and the DG-algebra Cob w ′′ ( D ′ ⊕ D ′′ ) is quasi-isomorphic to ob w ( D ′′ ). At the same time, the DG-algebras Cob w ′ ( D ′ ⊕ D ′′ ) and Cob w ′′ ( D ′ ⊕ D ′′ )are isomorphic as CDG-algebras over k . Remark.
One can call a strictly coaugmented strictly counital curved A ∞ -coalgebra C over a field k conilpotent if the DG-algebra A = L n C + [ − ⊗ n is cofibrant. Unlikecomodules over a conilpotent coalgebra, which are all conilpotent in the appropri-ate sense, not every strictly counital curved A ∞ -comodule over a conilpotent curvedA ∞ -coalgebra is conilpotent. For any strictly counital curved A ∞ -coalgebra C , con-sider the CDG-coalgebra U = Bar v ( A ); it can be called the conilpotent coenvelop-ing CDG-coalgebra of C . The DG-category of conilpotent curved A ∞ -comodulesover C and strict morphisms between them can be defined as the DG-category ofCDG-comodules over U . The faithful DG-functor N A ⊗ τ A,v N provides anembedding of the DG-category of conilpotent curved A ∞ -comodules over C andstrict morphisms between them into the DG-category of strictly counital curvedA ∞ -comodules over C . Analogously, the DG-category of contranilpotent curvedA ∞ -contramodules over C and strict morphisms between them can be defined as theDG-category of CDG-contramodules over U . The DG-functor Q Hom τ A,v ( A, Q )maps the DG-category of contranilpotent curved A ∞ -contramodules over C and strictmorphisms between them to the DG-category of strictly counital curved A ∞ -contra-modules. By Theorem 6.3 and the above Theorem, for any conilpotent curvedA ∞ -coalgebra C the coderived category of conilpotent curved A ∞ -comodules andstrict morphisms between them is equivalent to the coderived category of strictlycounital curved A ∞ -comodules, and the contraderived category of contranilpotentcurved A ∞ -contramodules and strict morphisms between them is equivalent to thecontraderived category of strictly counital curved A ∞ -contramodules. Appendix A. Homogeneous Koszul Duality
A.1.
Covariant homogeneous duality.
We will consider DG-algebras, DG-coal-gebras, DG-modules, DG-comodules, DG-contramodules endowed with an additional Z -valued grading, which will be called the internal grading and denoted by lowerindices. The internal grading is always assumed to be preserved by the differentials, d ( X n ) ⊂ X n . All morphisms of internally graded objects are presumed to preservethe internal gradings. The other grading, raised by 1 by the differentials, is alsoalways present; it is called the cohomological grading and denoted by upper indices,as above in this paper. We will always use the cohomological grading only in all thesign rules, so the notation | z | is understood to refer to the cohomological grading.The same applies to the notation V V [1], which is interpreted as shifting thecohomological grading and leaving the internal grading unchanged. et k be a field, and let A be a DG-algebra and C be a DG-coalgebra over k endowed with internal gradings. Assume that the bigraded vector spaces A + = A/k and C + = ker( C → k ) are concentrated in the positive internal grading, i. e., A + = L n> A n and C + = L n> C n , where A n and C n are complexes of vector spaces.Then A admits a unique augmentation v : A −→ k preserving the internal gradingand C admits a unique coaugmentation w : k −→ C with the same property. Noticethat any positively internally graded DG-coalgebra C is conilpotent. The bar andcobar-constructions Bar v ( A ) and Cob w ( C ) from 6.1 produce a DG-coalgebra and aDG-algebra endowed with internal and cohomological gradings.We will presume twisting cochains τ : C −→ A (see 6.2) to have the internaldegree 0 and the cohomological degree 1; besides, τ is assumed to satisfy theequations v ◦ τ = τ ◦ w = 0. These conditions hold for the natural twistingcochains τ A,v : Bar v ( A ) −→ A and τ C,w : C −→ Cob w ( C ). There are natural bi-jective correspondences between the three sets: the set of all morphisms of internallygraded DG-algebras Cob w ( C ) −→ A , the set of all morphisms of internally gradedDG-coalgebras C −→ Bar v ( A ), and the set of all twisting cochains τ : C −→ A asabove. So the functor Cob w is left adjoint to the functor Bar v . Theorem 1.
Let τ be a twisting cochain between a (positively internally graded)DG-coalgebra C and DG-algebra A . Then the following conditions are equivalent: (a) the morphism of DG-algebras Cob w ( C ) −→ A is a quasi-isomorphism; (b) the morphism of DG-coalgebras C −→ Bar v ( A ) is a quasi-isomorphism; (c) the complex A ⊗ τ C is quasi-isomorphic to k ; (d) the complex C ⊗ τ A is quasi-isomorphic to k .Besides, the functors Bar v and Cob w transform quasi-isomorphisms ofDG-algebras to quasi-isomorphisms of DG-coalgebras and vice versa. Conse-quently, these functors induce an equivalence between the categories of DG-algebrasand DG-coalgebras with inverted quasi-isomorphisms.Proof. The complex A ⊗ τ A,v
Bar v ( A ) is the reduced bar resolution of the leftDG-module k over A , so the standard contracting homotopy induced by the unitelement of A proves its acyclicity. The same argument applies to the complexesBar v ( A ) ⊗ τ A,v A , C ⊗ τ C,w
Cob w ( C ), and Cob w ( C ) ⊗ τ C,w C . If ( C ′ , A ′ , τ ′ ) −→ ( C ′′ , A ′′ , τ ′′ )is a morphism of twisting cochains for which the morphism of DG-coalgebras C ′ −→ C ′′ and the morphism of DG-algebras A ′ −→ A ′′ are quasi-isomorphisms, then theinduced morphisms of complexes A ′ ⊗ τ ′ C ′ −→ A ′′ ⊗ τ ′′ C ′′ and C ′ ⊗ τ ′ A ′ −→ C ′′ ⊗ τ ′′ A ′′ are quasi-isomorphisms. Indeed, every internal degree component of A ⊗ τ C and C ⊗ τ A is obtained from the tensor products of the internal degree components of A and C by a finite number of shifts and cones, and the operations of tensor product,shift, and cone preserve quasi-isomorphisms of complexes of vector spaces. A sim-ilar argument proves that the functors Bar v and Cob w map quasi-isomorphisms to uasi-isomorphisms, since the internal degree components of Bar v ( A ) and Cob w ( C )are obtained from the internal degree components of A and C by finite iterations oftensor products, shifts, and cones. Finally, one shows by induction in the internaldegree that if a morphism of twisting cochains ( C, A ′ , τ ′ ) −→ ( C, A ′′ , τ ′′ ) induces aquasi-isomorphism of complexes A ′ ⊗ τ ′ C −→ A ′′ ⊗ τ ′′ C or C ⊗ τ ′ A ′ −→ C ⊗ τ ′′ A ′′ thenthe morphism of A ′ −→ A ′′ is a quasi-isomorphism, and analogously for morphismsof twisting cochains ( C ′ , A, τ ′ ) −→ ( C ′′ , A, τ ′′ ). (cid:3) As in 6.5, a twisting cochain τ satisfying the equivalent conditions of Theorem 1 issaid to be acyclic . Any twisting cochain τ induces two pairs of adjoint functors be-tween the homotopy categories of internally graded DG-modules, DG-comodules,and DG-contramodules Hot ( A – mod ), Hot ( C – comod ), and Hot ( C – contra ). Thefunctor M C ⊗ τ M : Hot ( A – mod ) −→ Hot ( C – comod ) is right adjoint tothe functor N A ⊗ τ N : Hot ( C – comod ) −→ Hot ( A – mod ) and the functor P Hom τ ( C, P ) :
Hot ( A – mod ) −→ Hot ( C – contra ) is left adjoint to the functor Q Hom τ ( A, Q ) :
Hot ( C – contra ) −→ Hot ( A – mod ).Let Hot ( A – mod ↑ ) and Hot ( A – mod ↓ ) denote the homotopy categories of (inter-nally graded) left DG-modules over A concentrated in nonnegative and nonposi-tive internal degrees, respectively. Let Hot ( C – comod ↑ ) denote the homotopy cat-egory of left DG-comodules over C concentrated in nonnegative internal degreesand Hot ( C – contra ↓ ) denote the homotopy category of left DG-contramodules over C concentrated in nonpositive internal degrees. Let D ( A – mod ↑ ), D ( A – mod ↓ ), D ( C – comod ↑ ), and D ( C – contra ↓ ) denote the corresponding derived categories, i. e.,the triangulated categories obtained from the respective homotopy categories by in-verting the quasi-isomorphisms. Theorem 2.
Let τ : C −→ A be an acyclic twisting cochain. Then (a) The functors C ⊗ τ − : Hot ( A – mod ↑ ) −→ Hot ( C – comod ↑ ) and A ⊗ τ − : Hot ( C – comod ↑ ) −→ Hot ( A – mod ↑ ) induce mutually inverse equivalences between thederived categories D ( A – mod ↑ ) and D ( C – comod ↑ ) . (b) The functors
Hom τ ( C, − ) : Hot ( A – mod ↓ ) −→ Hot ( C – contra ↓ ) and Hom τ ( A, − ) : Hot ( C – contra ↓ ) −→ Hot ( A – mod ↓ ) induce mutually inverse equivalences be-tween the derived categories D ( A – mod ↓ ) and D ( C – contra ↓ ) .Proof. One can deduce this Theorem from Theorem 6.5, using the facts that anyacyclic DG-comodule from
Hot ( C – comod ↑ ) is coacyclic and any acyclic DG-contra-module from Hot ( C – contra ↓ ) is contraacyclic. Indeed, any nonnegatively internallygraded DG-comodule N over C is the inductive limit of DG-comodules obtained byshifts and cones from the internal degree components N n considered as DG-comoduleswith the trivial comodule structure. Analogously, any nonpositively internally raded DG-contramodule Q over C is the projective limit of DG-contramodules ob-tained by shifts and cones from the internal degree components Q n considered asDG-contramodules with the trivial contramodule structure. (Similarly one can showthat any nonnegatively internally graded DG-module over A that is projective as abigraded A -module is projective in the sense of 1.4 and any nonpositively internallygraded DG-module over A that is injective as a bigraded A -module is injective inthe sense of 1.5.) Alternatively, here is a direct proof.The functors M C ⊗ τ M and N A ⊗ τ N map acyclic DG-modules to acyclicDG-comodules and vice versa, since the internal degree components of C ⊗ τ M and A ⊗ τ N are obtained from the internal degree components of M and N by tensoringthem with the internal degree components of C and A and taking shifts and cones afinite number of times for each of the components of C ⊗ τ M and A ⊗ τ N . To checkthat the adjunction morphism A ⊗ τ C ⊗ τ M −→ M is a quasi-isomorphism, noticethat the integral degree components of its kernel can be obtained from the positiveinternal degree components of A ⊗ τ C by tensoring them with the internal degreecomponents of M and taking shifts and cones a finite number of times. Analogouslyone shows that the adjunction morphism N −→ C ⊗ τ A ⊗ τ N is a quasi-isomorphism.The proof of part (b) is similar, with the only difference that one has to use exactnessof the functor Hom k instead of ⊗ k . (cid:3) The mutually inverse functors C ⊗ τ − and A ⊗ τ − between the derived categories D ( A – mod ↑ ) and D ( C – comod ↑ ) transform DG-modules with the trivial action of A into cofree DG-comodules over C , free DG-modules over A into DG-comodules withthe trivial action of C . The mutually inverse functors Hom τ ( C, − ) and Hom τ ( A, − )between the derived categories D ( A – mod ↓ ) and D ( C – contra ↓ ) transform DG-moduleswith the trivial action of A into free DG-contramodules over C , cofree DG-modulesover A into DG-contramodules with the trivial action of C .As explained in 6.5, for any DG-module M ∈ D ( A – mod ↑ ) the complex C ⊗ τ M computes Tor A ( k, M ) and for any DG-comodule N ∈ D ( C – comod ↑ ) the complex A ⊗ τ N computes Cotor C ( k, N ) ≃ Ext C ( k, N ). For any DG-module P ∈ D ( A – mod ↓ )the complex Hom τ ( C, P ) computes Ext A ( k, P ) and for any DG-contramodule Q ∈ D ( C – contra ↓ ) the complex Hom τ ( A, Q ) computes Coext C ( k, Q ) ≃ Ctrtor C ( k, Q ).The DG-coalgebra C itself, considered as a complex, computes Tor A ( k, k ), and theDG-algebra A , considered as a complex, computes Cotor C ( k, k ) ≃ Ext C ( k, k ). Corollary.
Let f : A ′ −→ A ′′ be a morphism of DG-algebras and g : C ′ −→ C ′′ be a morphism of DG-coalgebras. Then the functor I R f : D ( A ′′ – mod ↑ ) −→ D ( A ′ – mod ↑ ) is an equivalence of triangulated categories if and only if the func-tor I R f : D ( A ′′ – mod ↓ ) −→ D ( A ′ – mod ↓ ) is an equivalence of triangulated categoriesand if and only if f is a quasi-isomorphism. The functor I R g : D ( C ′ – comod ↑ ) −→ D ( C ′′ – comod ↑ ) is an equivalence of triangulated categories if and only if the functor R g : D ( C ′ – contra ↓ ) −→ D ( C ′′ – contra ↓ ) is an equivalence of triangulated categoriesand if and only if g is a quasi-isomorphism.Proof. The facts that whenever f or g is a quasi-isomorphism the related restriction-of-scalars functors are equivalences of triangulated categories can be deduced fromTheorems 1.7 and 4.8, since any quasi-isomorphism of positively internally gradedDG-coalgebras is a filtered quasi-isomorphism. To prove the inverse implications, itsuffices to apply the adjoint functors L E f , R E f , R E g , and L E g to the DG-modules A and Hom k ( A, k ), the DG-comodule C , and the DG-contramodule Hom k ( C, k ). Allassertions of Corollary also follow from Theorems 1 and 2. (cid:3)
Let A = L ∞ n =0 A n and C = L ∞ n =0 C n be a Koszul graded algebra and a Koszulgraded coalgebra quadratic dual to each other [47]. Let C be the internally gradedDG-coalgebra with the components C jn = 0 for all j = 0 and C n = C n , and let A be the internally graded DG-algebra with the components A jn = 0 for j = n and A nn = A n , the multiplication in A and the comultiplication in C being given by themultiplication in A and the comultiplication in C , and the differentials in A and C being zero. Let τ : C −→ A be the twisting cochain that is given by the isomorphism C ≃ A in the internal degree 1 and vanishes in all the other internal degrees. Bythe definition of quadratic duality and Koszulity, τ is acyclic.The homotopy category Hot ( C – comod ) and the derived category D ( C – comod ) canbe identified in an obvious way with the homotopy category Hot ( C – comod ) and thederived category D ( C – comod ) of the abelian category of graded C -comodules. Anal-ogously, the homotopy category Hot ( C – contra ) and the derived category D ( C – contra )can be identified with the homotopy category Hot ( C – contra ) and the derived category D ( C – contra ) of the abelian category of graded C -contramodules. Finally, an obvioustransformation of the bigrading allows to identify the homotopy category Hot ( A – mod )and the derived category D ( A – mod ) with the homotopy category Hot ( A – mod ) andthe derived category D ( A – mod ) of the abelian category of graded A -modules.Hence we obtain the equivalences of derived categories D ( A – mod ↑ ) ≃ D ( C – comod ↑ )and D ( A – mod ↓ ) ≃ D ( C – contra ↓ ), where A – mod ↑ , A – mod ↓ , C – comod ↑ , and C – contra ↓ denote the abelian categories of nonnegatively graded A -modules, non-positively graded A -modules, nonnegatively graded C -comodules, and nonpositivelygraded C -contramodules, respectively. Notice that these are unbounded derived cat-egories of the abelian categories of the categories of graded modules, comodules, andcontramodules satisfying no finiteness conditions. The only restriction imposed onthe complexes of modules, comodules, and contramodules is that of the nonpositivityor nonnegativity of the internal grading. Remark.
The results of Theorems 1–2 can be generalized to the following situa-tion. Let E be an (associative, noncommutative) tensor DG-category (with shifts nd cones) and A ⊂ H ( E ) be a full triangulated subcategory that is a two-sidedtensor ideal. Consider graded algebra objects A and graded coalgebra objects C over E such that A n = 0 = C n for n < A and C are the unit object of E . Letus call a morphism of algebras A ′ −→ A ′′ or a morphism of coalgebras C ′ −→ C ′′ a quasi-isomorphism if its cone belongs to A in every degree. Then the analogue ofTheorem 1 holds for such graded algebras and coalgebras. Furthermore, let M be aleft module DG-category (with shifts and cones) over the tensor DG-category E andlet B ⊂ H ( M ) be a full triangulated subcategory such that H ( E ) ⊗ B ⊂ B and A ⊗ H ( M ) ⊂ B . Consider nonnegatively graded module objects M and comoduleobjects N in M over the graded algebras A and graded coalgebras C . Let us call amorphism of modules M ′ −→ M ′′ or a morphism of comodules N ′ −→ N ′′ a quasi-isomorphism if its cone belongs to B in every degree. Then the analogue of Theorem 2holds for such graded modules and comodules. In particular, one can take E to bethe tensor DG-category of DG-bimodules over a DG-ring R that are flat as rightDG-modules, M to be the left module DG-category of left DG-modules over R , and A and B to be the triangulated subcategories of acyclic DG-(bi)modules. One canalso take E to be the tensor DG-category of CDG-bicomodules over a CDG-coalgebra C that are injective as right graded comodules, M to be the left module DG-categoryof left CDG-comodules over C , and A and B to be the triangulated subcategories ofcoacyclic CDG-(bi)comodules. One can also use right module DG-categories oppositeto the DG-categories of left DG-modules or left CDG-contramodules with respect tothe action functors Hom R or Cohom C , assuming DG-bimodules to be projective asleft DG-modules or CDG-bicomodules to be injective as left graded comodules.A.2. Contravariant duality.
Let R be a (noncommutative) ring and let A and B be two internally graded DG-rings such that A n = 0 = B − n for n < A j = 0 = B j for j = 0, and A = R = B . Assume further that all the internal degree compo-nents A n are bounded complexes of finitely generated projective left R -modules andall the components B n are bounded complexes of finitely generated projective right R -modules. We will consider nonnegatively internally graded left DG-modules M over A and nonpositively internally graded right DG-modules N over B , and we willassume all their internal degree components M n and N n to be bounded complexesof finitely generated projective R -modules. Let Hot ( A – mod ↑ ) and Hot ( mod ↓ – B ) de-note the homotopy categories of such DG-modules, and D ( A – mod ↑ ) and D ( mod ↓ – B )denote the corresponding derived categories.Given two internally graded DG-rings A and B as above, construct two op-posite DG-rings T l and T r in the following way. Let T l = T r be the sub-complex in Q n B − n ⊗ R A n consisting of all the elements t satisfying the equa-tion rt = tr for all r ∈ R . Define the multiplication in T l by the formula( b ′ ⊗ a ′ )( b ′′ ⊗ a ′′ ) = ( − | a ′ | ( | b ′′ | + | a ′′ | ) b ′ b ′′ ⊗ a ′′ a ′ and the multiplication in T r by the ormula ( b ′ ⊗ a ′ )( b ′′ ⊗ a ′′ ) = ( − | b ′′ | ( | b ′ | + | a ′ | ) b ′′ b ′ ⊗ a ′ a ′′ . It is easy to see that thesemultiplications are well-defined, associative and compatible with the differential, andthat T l and T r are two opposite DG-rings in the sense of 3.12.By a twisting cochain for the DG-rings A and B we mean an element τ ∈ Q ∞ n =1 B − n ⊗ R A n of (cohomological) degree 1 satisfying the equation τ + dτ = 0in T l , or equivalently, the equation τ − dτ = 0 in T r .For a DG-ring A , consider the internally and cohomologically graded R - R -bimoduleHom R ( A + , R )[ − A + = A/R . There in a natural “trace” element τ A ∈ Q ∞ n =1 Hom R ( A n , R ) ⊗ R A n . Consider the ring B = Cob l ( A ) freely generated bythe bimodule Hom R ( A + , R )[ −
1] over R . Then τ A is an element of degree 1 in thecohomologically graded ring T l . Endow B = Cob l ( A ) with the unique differentialfor which τ A + dτ A = 0 in T l . Then B is an internally graded DG-ring and τ A is atwisting cochain for A and B . Analogously, for a DG-ring B consider the internallyand cohomologically graded R - R -bimodule Hom R op ( B + , R )[ − B + = B/R .There is a natural “trace” element τ B ∈ Q ∞ n =1 B − n ⊗ R Hom R op ( B − n , R ). Considerthe ring A = Cob r ( B ) freely generated by the bimodule Hom R op ( B + , R )[ −
1] over R .Then τ B is an element of degree 1 in the cohomologically graded ring T l . Endow A = Cob r ( B ) with the unique differential for which τ B + dτ B = 0 in T l . Then A isan internally graded DG-ring and τ A is a twisting cochain for A and B .There are natural bijective correspodences between three sets: the set of morphismsof internally graded DG-rings Cob l ( A ) −→ B , the set of morphisms of internallygraded DG-rings Cob r ( B ) −→ A , and the set of all twisting cochains τ for A and B .So the contravariant functors Cob l and Cob r are left adjoint to each other.Let τ be a twisting cochain for A and B . Given a DG-module M over A , constructa DG-module Hom τ ( M, B ) over B in the following way. As an internally and coho-mologically graded right B -module, Hom τ ( M, B ) coincides with Hom R ( M, B ), wherethe action of B is given by the formula ( f b )( x ) = ( − | b || x | f ( x ) b . On the complexHom R ( M, B ) there is a structure of left DG-module over the DG-ring T l given bythe formula ( b ⊗ a )( f )( x ) = ( − | a || f | bf ( ax ) for x ∈ M and f ∈ Hom R ( M, B ). Theleft action of T l on Hom R ( M, B ) commutes with the right action of B . Thus one cantwist the differential d on Hom R ( M, B ) by replacing it with d + τ , where τ acts onHom R ( M, B ) as an element of T l . So we obtain the DG-module Hom τ ( M, B ) over B .Given a DG-module N over B , construct a DG-module Hom( N, A ) τ over A asfollows. As an internally and cohomologically graded left A -module, Hom( N, A ) τ coincides with Hom R op ( N, A ). On the complex Hom R op ( N, A ) there is a struc-ture of left DG-module over the DG-ring T l given by the formula ( b ⊗ a )( g )( y ) =( − ( | b | + | a | )( | g | + | y | ) g ( yb ) a for y ∈ N and g ∈ Hom R op ( N, A ). The left action of T l on Hom R op ( N, A ) supercommutes with the left action of A . Thus one can twist thedifferential d on Hom R op ( N, A ) by replacing it with d + τ , where τ acts as an elementof T l . So we obtain the DG-module Hom( N, A ) τ over A . heorem 1. Let τ be a twisting cochain between internally graded DG-rings A and B .Then the following conditions are equivalent: (a) the morphism of DG-rings Cob l ( A ) −→ B is a quasi-isomorphism; (b) the morphism of DG-rings Cob r ( B ) −→ A is a quasi-isomorphism; (c) the complex Hom τ ( A, B ) is quasi-isomorphic to R ; (d) the complex Hom(
B, A ) τ is quasi-isomorphic to R .Besides, the functors Cob l and Cob r transform quasi-isomorphisms of inter-nally graded DG-rings to quasi-isomorphisms. Consequently, they induce an anti-equivalence between the categories of DG-rings of the A and B kind with invertedquasi-isomorphisms.Proof. The complex Hom τ A ( A, Cob l ( A )) is the reduced cobar-resolution of the leftDG-module R over A relative to R , and the complex Hom(Cob l ( A ) , A ) τ A is thereduced bar-resolution of the left DG-module R over A relative to R . So thesecomplexes are quasi-isomorphic to R . The rest of the proof is analogous to that ofTheorem A.1.1. It uses the exactness and reflexivity properties of the tensor product,Hom, and duality for bounded complexes of finitely generated projective modules. (cid:3) A twisting cochain τ satisfying the equivalent conditions of Theorem 1 is saidto be acyclic . Any twisting cochain τ induces a pair of left adjoint contravari-ant functors M Hom τ ( M, B ) :
Hot ( A – mod ↑ ) op −→ Hot ( mod ↓ – B ) and N Hom(
N, A ) τ : Hot ( mod ↓ – B ) op −→ Hot ( A – mod ↑ ). Theorem 2.
Let τ be an acyclic twisting cochain for internally graded DG-rings A and B . Then the functors Hom τ ( − , B ) and Hom( − , A ) τ induce mutually inverseanti-equivalences between the derived categories D ( A – mod ↑ ) and D ( mod ↓ – B ) .Proof. The functor Hom τ ( − , B ) preserves acyclicity of DG-modules, since the in-ternal degree components of Hom τ ( M, B ) are obtained from the internal degreecomponents of M by applying the functor Hom R into the internal degree compo-nents of B and taking shifts and cones. To check that the adjunction morphismHom(Hom τ ( M, B ) , A ) τ −→ M is a quasi-isomorphism, notice that the internal degreecomponents of its kernel can be obtained from the positive internal degree compo-nents of Hom( B, A ) τ by applying the functor ⊗ R with the internal degree componentsof M and taking shifts and cones.In fact, stronger assertions hold: every DG-module in Hot ( A – mod ↑ ) that is pro-jective as a graded A -module is projective in the sense of 1.4, and consequently thefunctor Hom τ ( − , B ) maps acyclic DG-modules to contractible ones. (cid:3) The mutually inverse functors Hom τ ( − , B ) and Hom( − , A ) τ transform the trivialDG-module R over A into the free DG-module B over B and the free DG-module A over A into the trivial DG-module R over B . or any DG-module M over A , the complex Hom τ ( M, B ) computes Ext A ( M, R ).Indeed, Hom(Hom τ ( M, B ) , A ) τ is a projective DG-module over A quasi-isomorphicto M , and Hom τ ( M, B ) ≃ Hom A (Hom(Hom τ ( M, B ) , A ) τ , R ). Analogously, forany DG-module N over B , the complex Hom( N, A ) τ computes Ext B op ( N, R ). TheDG-ring B itself computes the Yoneda Ext-ring Ext A ( R, R ) op , and the DG-ring A computes Ext B ( R, R ). Corollary.
Let f : A ′ −→ A ′′ be a morphism of nonnegatively internally gradedDG-rings as above. Then the functor I R f : D ( A ′′ – mod ↑ ) −→ D ( A ′ – mod ↑ ) is an equiv-alence of triangulated categories if and only if f is a quasi-isomorphism.Proof. This can be deduced either from Theorem 1.7, or from Theorems 1 and 2. (cid:3)
Remark.
In the case when R has a finite (left and right) homological dimension thereare several other versions of the above duality. E. g., one can consider DG-rings A and B whose internal degree components A n and B n are bounded complexes of finitelygenerated projective R -modules, and DG-modules M and N whose components M n and N n are unbounded complexes of finitely generated projective R -modules. Onecan also consider DG-rings A and DG-modules M whose components are complexesof finitely generated projective left R -modules bounded from above, and DG-rings B and DG-modules N whose components are complexes of finitely generated projectiveright R -modules bounded from below. The above Theorems 1 and 2 hold in thesesituations. Assuming additionally that R is (left and right) Noetherian one can evendrop the projectivity assumption on the bigrading components of the modules in all ofthese results. This is so because the derived categories of modules whose components M n and N n are (appropriately bounded) complexes of finitely generated R -modulesare equivalent to the derived categories of modules whose components are (accord-ingly bounded) complexes of finitely generated projective R -modules. (Cf. [38].) Appendix B. D – Ω Duality
B.1.
Duality functors.
Let X be a scheme; denote by O X its structure sheaf. A quasi-coherent CDG-algebra B over X is a graded quasi-coherent O X -algebra B = L i B i endowed with the following structure. For each affine open subscheme U ⊂ X there is a structure of CDG-ring on the graded ring B ( U ), i. e., a (not necessarily O X -linear) odd derivation d : B ( U ) −→ B ( U ) of degree 1 and an element h ∈ B ( U )satisfying the usual equations are given. For each pair of embedded affine subschemes U ⊂ V ⊂ X an element a UV ∈ B ( U ) is fixed such that the pair ( ρ UV , a UV ) is amorphism of CDG-rings B ( V ) −→ B ( U ), where ρ UV is the restriction map in thesheaf B . For any three embedded affine subschemes U ⊂ V ⊂ W ⊂ X , the morphism ρ UW , a UW ) is equal to the composition of the morphisms ( ρ V W , a
V W ) and ( ρ UV , a UV ),i. e., the equation a UW = a UV + ρ UV ( a V W ) holds.A morphism of quasi-coherent CDG-algebras
B −→ A over X is a morphism ofgraded quasi-coherent O X -algebras together with a family of morphisms of CDG-rings B ( U ) −→ A ( U ) defined for all affine open subschemes U ⊂ X and satisfying the obvi-ous compatibility conditions. A quasi-coherent (left or right) CDG-module N over aquasi-coherent CDG-algebra B is an O X -quasi-coherent sheaf of graded modules over B together with a family of differentials d : N ( U ) −→ N ( U ) of degree 1 on the graded B ( U )-modules N ( U ) defined for all affine open subschemes U ⊂ X which make N ( U )into CDG-modules over the CDG-rings B ( U ) and satisfy the obvious compatibilitieswith respect to the restriction morphisms N ( V ) −→ N ( U ) and B ( V ) −→ B ( U ).Quasi-coherent left CDG-modules over a fixed quasi-coherent CDG-algebra B forma DG-category denoted by DG ( B – mod ) and quasi-coherent right CDG-modules formthe DG-category DG ( mod – B ); the corresponding homotopy categories are denotedby Hot ( B – mod ) and Hot ( mod – B ). When X is affine, the category of quasi-coherentCDG-algebras B over X is equivalent to the category of CDG-rings ( B, d, h ) for whichthe graded ring B is an O ( X )-algebra; the categories DG ( B – mod ) and DG ( mod – B )are equivalent to the DG-categories DG ( B – mod ) and DG ( mod – B ).The coderived categories D co ( B – mod ) and D co ( mod – B ) are defined in the usualway as the quotient categories of the homotopy categories Hot ( B – mod ) and Hot ( mod – B ) by their minimal triangulated subcategories containing the total quasi-coherent CDG-modules of the exact triples in the abelian category of quasi-coherentCDG-modules (with closed morphisms between them) and closed under infinitedirect sums. Notice that we do not define contraderived categories of quasi-coherentCDG-modules, because infinite products of quasi-coherent sheaves are not well-behaved. A reader who finds the above definitions too sketchy may wish to consultSection 3 for a detailed discussion of the affine case.From now on, let X be a smooth algebraic variety (a smooth separated scheme offinite type) over a field k . Let E be an algebraic vector bundle (locally free sheaf)over X . We denote by D X, E the sheaf of differential operators on X acting in thesections of E . These words have a unique meaning in the case of a field k of charac-teristic 0, and in the finite characteristic case we are interested in the “crystalline”differential operators. More precisely, let E nd ( E ) be the sheaf of endomorphisms of E and F D X, E be the sheaf of differential operators of order at most 1 acting in thesections of E . Then we define D X, E as the sheaf of rings on X generated by the bi-module F D X, E ⊃ E nd ( E ) over the quasi-coherent sheaf of O X -algebras E nd ( E ) withthe relations uv − vu = [ u, v ] for any two local sections u and v of F D X, E such thatthe image of at least one of them in F D X,E / E nd ( E ) ≃ E nd ( E ) ⊗ O X T X belongs to T X ⊂ E nd ( E ) ⊗ O X T X , where T X denotes the sheaf of vector fields on X . ssume that X is affine. Then there exists a global connection ∇ E in the vec-tor bundle E . Denote by End( E ) and D X, E the global section rings of the sheavesof rings E nd ( E ) and D X, E on X . Let Ω( X, End( E )) = Ω( X ) ⊗ O ( X ) End( E ) denotethe graded ring of global differential forms on X with coefficients in End( E ). Theconnection ∇ E induces a connection ∇ E nd ( E ) in the vector bundle E nd ( E ), so thereis the de Rham differential d on Ω( X, End( E )) related to the connection ∇ E nd ( E ) .Let h ∈ Ω ( X, End( E )) be the curvature of the connection ∇ E . Then the differen-tial d and the curvature element h define a structure of CDG-ring on the gradedring Ω( X, End( E )). The de Rham differential on the graded Ω( X, End( E ))-module K ( X, E ) = Ω( X, End( E )) ⊗ End( E ) D X, E ≃ Ω( X ) ⊗ O ( X ) D X, E makes it a leftCDG-module over Ω( X, End( E )) with a commuting structure of a right D X, E -module.Now let X be an arbitrary smooth algebraic variety. Denote by Ω X ( E nd ( E )) =Ω X ⊗ O X E nd ( E ) the sheaf of differential forms on X with coefficients in the vec-tor bundle E nd ( E ); it is a graded quasi-coherent O X -algebra. Choosing connec-tions ∇ E| U on the restrictions of E on all the affine open subschemes U ⊂ X , onecan define a structure of quasi-coherent CDG-algebra on Ω X ( E nd ( E )). The gradedΩ X ( E nd ( E ))-module K ( X, E ) = Ω X ( E nd ( E )) ⊗ E nd ( E ) D X, E ≃ Ω X ⊗ O X D X, E becomesa quasi-coherent left CDG-module over Ω X ( E nd ( E )) with a commuting structure ofa (quasi-coherent) right D X, E -module.To any complex of right D X, E -modules M one can assign the quasi-coherentright CDG-module H om D op X, E ( K ( X, E ) , M ) ≃ H om E nd ( E ) op (Ω X ( E nd ( E )) , M ) overΩ X ( E nd ( E )). Conversely, to any quasi-coherent right CDG-module N over Ω X ( E nd ( E )) one can assign the complex of right D X, E -modules N ⊗ Ω X ( E nd ( E )) K ( X, E ) ≃N ⊗ E nd ( E ) D X, E . This defines a pair of adjoint functors between the homotopycategories Hot ( mod –Ω X ( E nd ( E ))) −→ Hot ( mod – D X, E ) and Hot ( mod – D X, E ) −→ Hot ( mod –Ω X ( E nd ( E ))).Whenever X is affine, there is also a pair of adjoint functors between the leftmodule categories. To any complex of left D X, E -modules P one can assign the leftCDG-module K ( X, E ) ⊗ D X, E P ≃ Ω( X, End( E )) ⊗ End( E ) P over Ω( X, End( E )). Con-versely, to any left CDG-module Q over Ω( X, End( E )) one can assign the complexof left D X, E -modules Hom Ω( X, End( E )) ( K ( X, E ) , Q ) ≃ Hom
End( E ) ( D X, E , Q ). This de-fines a pair of adjoint functors between the homotopy categories Hot ( D X, E – mod ) −→ Hot (Ω( X, End( E ))– mod ) and Hot (Ω( X, End( E ))– mod ) −→ Hot ( D X, E – mod ).B.2. Duality theorem.
Let X be a smooth algebraic variety over a field k and E be an algebraic vector bundle over X . Denote by Hot ( mod – D X, E ) and D ( mod – D X, E )the (unbounded) homotopy and derived categories of the abelian category of (quasi-coherent) right D X, E -modules. Theorem. (a)
The adjoint functors H om E nd ( E ) op (Ω X ( E nd ( E )) , − ) : Hot ( mod – D X, E ) −→ Hot ( mod –Ω X ( E nd ( E ))) and − ⊗ E nd ( E ) D X, E : Hot ( mod –Ω X ( E nd ( E ))) −→ ot ( mod – D X, E ) induce functors D ( mod – D X, E ) −→ D co ( mod –Ω X ( E nd ( E ))) and D co ( mod –Ω X ( E nd ( E ))) −→ D ( mod – D X, E ) , which are mutually inverse equivalencesof triangulated categories. (b) Assume that X is affine. Then the adjoint functors Ω( X, End( E )) ⊗ End( E ) − : Hot ( D X, E – mod ) −→ Hot (Ω( X, End( E ))– mod ) and Hom
End( E ) ( D X, E , − ) : Hot (Ω( X, End( E ))– mod ) −→ Hot ( D X, E – mod ) induce functors D ( D X, E – mod ) −→ D ctr (Ω( X, End( E ))– mod ) and D ctr (Ω( X, End( E ))– mod ) −→ D ( D X, E – mod ) , which aremutually inverse equivalences of triangulated categories.Proof. We will prove part (a); the proof of part (b) is analogous up to duality.For simplicity of notation, denote by P X, E the graded E nd ( E )- E nd ( E )-bimodule L i E nd ( E ) ⊗ O X Λ i T X of polyvector fields on X with coefficients in E nd ( E ); then H om E nd ( E ) op (Ω X ( E nd ( E )) , M ) ≃ M ⊗ E nd ( E ) P X, E for any right E nd ( E )-module M .We will make use of the decreasing filtration F of Ω X ( E nd ( E )) defined by therule F i Ω X ( E nd ( E )) = L j > i Ω jX ( E nd ( E )), the dual increasing filtration F of P X, E given by the rule F i P X, E = L j i P jX, E , where P jX, E = E nd ( E ) ⊗ O X Λ j T X , andthe increasing filtration F of the sheaf D X, E by the order of differential opera-tors. To prove that the induced functors exist, one can notice that the cate-gory of right D X, E -modules has a finite homological dimension, so D ( mod – D X, E ) ≃ D co ( mod – D X, E ), and the functors − ⊗ E nd ( E ) P X, E and − ⊗ E nd ( E ) D X, E transform ex-act triples of complexes of right D X, E -modules into exact triples of quasi-coherentright CDG-modules over Ω X ( E nd ( E )) and vice versa. Alternatively, for any com-plex of right D X, E -modules M consider the increasing filtration of the quasi-coherentCDG-module M ⊗ E nd ( E ) P X, E induced by the increasing filtration F of P X, E . It isa filtration by quasi-coherent CDG-submodules and the associated quotient quasi-coherent CDG-modules M ⊗ E nd ( E ) P iX, E only depend on the E nd ( E )-module struc-tures on the components of M , so it suffices to know that the category of right E nd ( E )-modules has a finite homological dimension and consequently any acycliccomplex of right E nd ( E )-modules is coacyclic.Furthermore, for any complex of right D X, E -modules M we have to show that themorphism of complexes of right D X, E -modules M⊗ E nd ( E ) P X, E ⊗ E nd ( E ) D X, E −→ M is aquasi-isomorphism. Consider the increasing filtration F of M⊗ E nd ( E ) P X, E ⊗ E nd ( E ) D X, E by complexes of E nd ( E )-submodules induced by the increasing filtrations F of P X, E and D X, E ; then the associated quotient complex of this filtration is the tensorproduct of the complex of right E nd ( E )-modules M with the Koszul complex of E nd ( E )- E nd ( E )-bimodules P X, E ⊗ E nd ( E ) gr F D X, E . Since the positive grading compone-nents of the Koszul complex are finite acyclic complexes of flat left E nd ( E )-modules,we are done. Finally, let N be a quasi-coherent right CDG-module over Ω X ( E nd ( E ));let us prove that the cone of the morphism of quasi-coherent right CDG-modules N −→ N ⊗ E nd ( E ) D X, E ⊗ E nd ( E ) P X, E is coacyclic. Let us reduce the question to he case when N is a complex of right E nd ( E )-modules with a trivial CDG-modulestructure, i. e., N · Ω iX ( E nd ( E )) = 0 for i >
0. Here it suffices to consider the finitedecreasing filtration G on N defined by the rules G i N = N · F i Ω X ( E nd ( E )) and theinduced filtration on N ⊗ E nd ( E ) D X, E ⊗ E nd ( E ) P X, E . Now when N is just a complex ofright E nd ( E )-modules, the increasing filtration F of N ⊗ E nd ( E ) D X, E ⊗ E nd ( E ) P X, E in-duced by the increasing filtrations F of D X, E and P X, E is a filtration by quasi-coherentCDG-submodules, the associated quotient quasi-coherent CDG-module is also just acomplex of right E nd ( E )-modules, and it remains to use the acyclicity and flatnessof the Koszul complex again. (cid:3) B.3.
Coderived and contraderived categories.
Let X be a smooth affine alge-braic variety of dimension d over a field k and E be an algebraic vector bundle over X .Consider also the algebraic vector bundle Ω d E ∗ = Ω dX ⊗ O X E ∗ over X , where E ∗ isthe dual vector bundle to E and Ω dX is the line bundle of differential forms of topdegree. It is well-known that the ring D op X, E opposite to D X, E is naturally isomorphicto D X, Ω d E ∗ , so right D X, Ω d E ∗ -modules can be considered as left D X, E -modules.On the other hand, there is a natural equivalence between the DG-categoryof right CDG-modules over the CDG-ring Ω( X, End(Ω d E ∗ )) corresponding to thevector bundle Ω d E ∗ and the DG-category left CDG-modules over the CDG-ringΩ( X, End( E )) corresponding to the vector bundle E . This equivalence assigns to aright CDG-module N over Ω( X, End(Ω d E ∗ )) the left CDG-module Ω d ( X ) ⊗ O ( X ) N over Ω( X, End( E )). The corresponding two equivalences of homotopy categoriestransform the functor Hom End(Ω d E ∗ ) op (Ω( X, End(Ω d E ∗ )) , − ) : Hot ( mod – D X, Ω d E ∗ ) −→ Hot ( mod –Ω( X, End(Ω d E ∗ ))) into the functor Ω( X, End( E )) ⊗ End( E ) − : Hot ( D X, E – mod ) −→ Hot (Ω( X, End( E ))– mod ).Thus by Theorem B.2 the same functor Ω( X, End( E )) ⊗ End( E ) − induces equiv-alences of the derived category D ( D X, E – mod ) with both the coderived category D co (Ω( X, End( E ))– mod ) and the contraderived category D ctr (Ω( X, End( E ))– mod ) ofCDG-modules. The next Theorem provides a more explicit equivalence betweenthese coderived and contraderived categories making a commutative diagram withthe above two equivalences. Theorem.
There is a natural equivalence of triangulated categories D co (Ω( X, End( E ))– mod ) ≃ D ctr (Ω( X, End( E ))– mod ) .Proof. The proof is similar to that of Theorem 3.9, with the following changes (cf. [48,Theorem 5.5]). In place of graded modules of finite injective (projective) dimension,one considers the graded modules over the graded ring Ω( X, End( E )) that are inducedfrom graded modules over End( E ), or equivalently, coinduced from graded modulesover End( E ). These are the graded modules of the form Ω( X, End( E )) ⊗ End( E ) P ,or equivalently, of the form Hom End( E ) (Ω( X, End( E )) , M ), where P and M aregraded left End( E )-modules. The quotient category of the homotopy category of eft CDG-modules over Ω( X, End( E )) whose underlying graded modules are induced(coinduced) from graded End( E )-modules by the minimal triangulated subcategorycontaining the exact triples of left CDG-modules that as exact triples of gradedmodules are induced (coinduced) from exact triples of graded End( E )-modules isequivalent to both D co (Ω( X, End( E ))– mod ) and D ctr (Ω( X, End( E ))– mod ). (cid:3) B.4.
Filtered D -modules. Let X be a smooth algebraic variety over a field k and E be an algebraic vector bundle over X . Let F denote the increasing filtration ofthe sheaf of differential operators D X, E by the order of differential operators. Con-sider the category mod ↑ fil – D X, E of right D X, E -modules M endowed with a cocompleteincreasing filtration F (by quasi-coherent O X -submodules) compatible with the fil-tration F on D X, E and such that F − M = 0. The category mod ↑ fil – D X, E has a naturalexact category structure; a triple of D X, E -modules is exact if and only if all its fil-tration components are exact triples of O X -modules. Denote by Hot ( mod ↑ fil – D X, E )and D ( mod ↑ fil – D X, E ) the (unbounded) homotopy and derived categories of this exactcategory. When X is affine, one can also consider the exact category D X, E – mod ↓ fil of left D X, E -modules endowed with a complete filtration F indexed by nonpositiveintegers, · · · ⊂ F − M ⊂ F − M ⊂ F M = M , and compatible with the filtration F on D X, E . The homotopy and derived categories of this exact category will be denotedby Hot ( D X, E – mod ↓ fil ) and D ( D X, E – mod ↓ fil ).Assuming that X is affine, consider the graded ring Ω( X, End( E )) ∼ constructedin 3.2 (now we are not interested in its DG-ring structure). For any X , the gradedrings Ω( U, End(
E | U )) for affine subschemes U ⊂ X glue together naturally forminga sheaf of graded rings Ω( X, E nd ( E )) ∼ over X . The sheaf Ω( X, E nd ( E )) ∼ is nonpos-itively graded and its zero-degree component is the sheaf of rings E nd ( E ). Noticethat Ω( X, E nd ( E )) ∼ is not a quasi-coherent O X -algebra, as the image of O X is notcontained in its center. The abelian category of quasi-coherent sheaves of graded(left or right) modules over the sheaf of graded rings Ω( X, E nd ( E )) ∼ is isomorphic tothe abelian category of quasi-coherent (left or right) CDG-modules over the quasi-coherent CDG-algebra Ω( X, E nd ( E )) and closed morphisms between them.We will find it convenient to consider Ω( X, E nd ( E )) ∼ as a sheaf of bigraded ringswith an internal and a cohomological grading, or a sheaf of internally graded DG-ringswith a zero differential, as explained in Appendix A. So we place Ω( X, E nd ( E )) ∼ inthe nonpositive internal grading n and the nonnegative cohomological grading i = − n running from i = 0 to i = dim X + 1. The obvious bigrading transformation identifiessheaves of internally graded DG-modules over Ω( X, E nd ( X )) ∼ considered as a sheafof internally graded DG-rings with complexes of sheaves of graded modules over thesheaf of graded rings Ω( X, E nd ( X )) ∼ . We will prefer the DG-module language.Denote by Hot ( mod ↑ –Ω( X, E nd ( E )) ∼ ) the homotopy category of sheaves of nonneg-atively internally graded O X -quasi-coherent right DG-modules over Ω( X, E nd ( E )) ∼ nd by D ( mod ↑ –Ω( X, E nd ( E )) ∼ ) the corresponding derived category, i. e., the quo-tient category of the homotopy category by the thick subcategory of sheaves ofDG-modules that are acyclic as complexes of sheaves in every internal grading. When X is affine, consider also the homotopy category Hot (Ω( X, End( E )) ∼ – mod ↓ ) of non-positively internally graded left DG-modules over the internally graded DG-ringΩ( X, End( E )) ∼ and the corresponding derived category D (Ω( X, End( E )) ∼ – mod ↓ ).These homotopy and derived categories are isomorphic to the (unbounded) homo-topy and derived categories of the corresponding abelian categories of quasi-coherent(nonnegatively) graded sheaves or (nonpositively) graded modules.The tensor product L ( X, E ) = Ω( X, E nd ( E )) ∼ ⊗ E nd ( E ) D X, E has a natural structureof a sheaf of (internally ungraded) left DG-modules over Ω( X, E nd ( E )) ∼ with the dif-ferential defined by the formula d L ( φ ⊗ p ) = ( − | φ | φd K (1 ⊗ p ) − ( − | φ | φδ ⊗ p , where d K denotes the differential of the CDG-module K ( X, E ) = Ω( X, E nd ( E )) ⊗ E nd ( E ) D X, E ,while φ ∈ Ω( X, E nd ( E )) ∼ and p ∈ D X, E . One defines the filtration F on L ( X, E )as the tensor product of the increasing filtration F on D X, E and the filtration F onΩ( X, E nd ( E )) ∼ indexed by nonpositive integers and associated with the internal grad-ing of Ω( X, E nd ( E )) ∼ . The latter filtration is given by the rule F j Ω( X, E nd ( E )) ∼ = L n j Ω( X, E nd ( E )) ∼ n , where n = − dim X −
1, . . . , 0. This endowes L ( X, E ) with astructure of a complex of filtered right D X, E -modules with the right action of D X, E commuting with the left action of Ω( X, E nd ( E )) ∼ . For X affine, we denote by L ( X, E )the DG-module of global sections of the sheaf L ( X, E ).Using the sheaf of DG-modules L ( X, E ), one can construct a pair of adjoint functorsbetween the homotopy categories Hot ( mod ↑ fil – D X, E ) and Hot ( mod ↑ –Ω( X, E nd ( E )) ∼ .To a complex of filtered right D X, E -modules M one assigns the sheaf of in-ternally graded right DG-modules H om L j F j D op X, E ( L j F j L ( X, E ) , L j F j M ) ≃H om E nd ( E ) op (Ω( X, E nd ( E )) ∼ , L j F j M ), where the direct sums of the filtrationcomponents L j F j M etc. are considered as (sheaves of) internally graded mod-ules in the internal grading j . To a sheaf of internally graded right DG-modules N over Ω( X, E nd ( E )) ∼ one assigns the complex of filtered right D X, E -modules( L n N n ) ⊗ Ω( X, E nd ( E )) L ( X, E ) ≃ ( L n N n ) ⊗ E nd ( E ) D X, E , where L n N is the notationfor the sheaf of internally ungraded DG-modules corresponding to N , and the tensorproducts are endowed with the filtration F induced by the filtration F on L ( X, E )or D X, E and the increasing filtration F on L n N corresponding to the grading of N .When X is affine, one can also use the DG-module L ( X, E ) to construct apair of adjoint functors between the homotopy categories Hot ( D X, E – mod ↓ fil ) and Hot (Ω( X, End( E )) ∼ – mod ↓ ). To a complex of filtered left D X, E -modules M one as-signs the internally graded left DG-module L j F j L ( X, E ) ⊗ L j F j D X, E L j M/F j − M ≃ Ω( X, End( E )) ∼ ⊗ End( E ) L j M/F j − M , where the direct sum of the filtration quotients j M/F j − M is considered as an internally graded L j F j D X, E -module with the in-ternal grading j . To an internally graded left DG-module N over Ω( X, End( E )) ∼ oneassigns the complex of filtered left D X, E -modules Hom Ω( X, End( E )) ∼ ( L ( X, E ) , Q n N n ) ≃ Hom
End( E ) ( D X, E , Q n N n ), where Q n N n is the internally ungraded DG-module con-structed by taking infinite products of components with a fixed cohomological grad-ing, and the modules Hom are endowed with the filtration F induced by the filtration F on L ( X, E ) or D X, E and the complete filtration F on Q n N n , indexed by nonpositiveintegers and coming from the grading of N . Theorem. (a)
The adjoint functors
M 7−→ H om E nd ( E ) op (Ω( X, E nd ( E )) ∼ , L j F j M ) and N 7−→ ( L n N n ) ⊗ E nd ( E ) D X, E between the homotopy categories Hot ( mod ↑ fil – D X, E ) and Hot ( mod ↑ –Ω( X, E nd ( E )) ∼ ) induce functors D ( mod ↑ fil – D X, E ) −→ D ( mod ↑ –Ω( X, E nd ( E )) ∼ ) and D ( mod ↑ –Ω( X, E nd ( E )) ∼ ) −→ D ( mod ↑ fil – D X, E ) , which aremutually inverse equivalences of triangulated categories. (b) Assume that X is affine. Then the adjoint functors M Ω( X, End( E )) ∼ ⊗ End( E ) L j M/F j − M and N Hom
End( E ) ( D X, E , Q n N n ) between the homo-topy categories Hot ( D X, E – mod ↓ fil ) and Hot (Ω( X, End( E )) ∼ – mod ↓ ) induce functors D ( D X, E – mod ↓ fil ) −→ D (Ω( X, End( E )) ∼ – mod ↓ ) and D (Ω( X, End( E )) ∼ – mod ↓ ) −→ D ( D X, E – mod ↓ fil ) , which are mutually inverse equivalences of triangulated categories.Proof. This is essentially a particular case of the generalization of Theorem A.1.2described in Remark A.1. First notice that the derived category of the exact cate-gory of filtered D X, E -modules with the filtrations indexed by nonnegative integersis equivalent to the derived category of the abelian category of (quasi-coherent)sheaves of nonnegatively graded modules over the sheaf of graded rings L j F j D X, E ,the equivalence being given by the functor M 7−→ L j F j M . Analogously, thederived category of the exact category of filtered D X, E -modules with the filtra-tions indexed by nonpositive integers is equivalent to the derived category of theabelian category of nonpositively graded modules over the graded ring L j F j D X, E ,the equivalence being given by the functor M L j M/F j − M . Furthermore, asin the proof of Theorem B.2, denote by P ∼ X, E the graded E nd ( E )- E nd ( E )-bimodule H om E nd ( E ) op (Ω( X, E nd ( E )) ∼ , E nd ( E )). To check that the components of positive in-ternal grading of the complexes P ∼ X, E ⊗ E nd ( E ) L j F j D X, E and ( L j F j D X, E ) ⊗ E nd ( E ) P ∼ X, E are acyclic, consider the decreasing filtrations G on these complexes induced by thefiltration G on L j F j D X, E given by the rule G t F j D X, E = F j − t D X, E for t > G on P ∼ X, E given by the rules G P ∼ X, E = P ∼ X, E , G P ∼ X, E = ker( P ∼ X, E → P X, E ).The associated graded complexes to these filtrations are the tensor products over k ofthe Koszul complexes P X, E ⊗ E nd ( E ) gr F D X, E and gr F D X, E ⊗ E nd ( E ) P X, E with the Koszulcomplex for the symmetric algebra and the exterior coalgebra in one variable. (Noticethat, in particular, the first-degree component of P ∼ X, E is isomorphic to F D ∼ X, E .) It emains to use the exactness of the tensor products of complexes of E nd ( E )-moduleswith finite complexes of flat E nd ( E )-modules and homomorphisms from finite com-plexes of projective End( E )-modules into complexes of End( E )-modules. (cid:3) Notice that one has D ( mod ↑ –Ω( X, E nd ( E )) ∼ ) = D co ( mod ↑ –Ω( X, E nd ( E )) ∼ ), i. e.,an O X -quasi-coherent sheaf of nonnegatively internally graded right DG-modulesover Ω( X, E nd ( E )) ∼ is coacyclic whenever it is acyclic. Analogously, one has D (Ω( X, End( E )) ∼ – mod ↓ ) = D ctr (Ω( X, End( E )) ∼ – mod ↓ ) when X is affine. One provesthis using the filtrations of (sheaves of) internally graded DG-modules associated withthe internal gradings, as explained in the proof of Theorem A.1.2, together with finite-ness of the homological dimension of the category of quasi-coherent sheaves over X .The functor of forgetting the filtration, acting from the derived category offiltered right D X, E -modules with filtrations indexed by nonnegative integers tothe derived category of right D X, E -modules, corresponds to the following functor D ( mod ↑ –Ω( X, E nd ( E )) ∼ ) −→ D co ( mod –Ω( X, E nd ( E ))). Given an O X -quasi-coherentsheaf N of nonnegatively graded right DG-modules over Ω( X, E nd ( E )) ∼ , one assignsthe quasi-coherent right CDG-module L n N n over Ω( X, E nd ( E )) with the actionof Ω( X, E nd ( E )) induced by the action of Ω( X, E nd ( E )) ∼ in N and the differen-tial d = δ + d ′ , where d ′ denotes the differential in the DG-module N . When X is affine, the functor of forgetting the filtration, acting from the derived cate-gory of filtered left D X, E -modules with filtrations indexed by nonpositive integersto the derived category of left D X, E -modules, corresponds to the following func-tor D (Ω( X, End( E )) ∼ – mod ↓ ) −→ D ctr (Ω( X, End( E ))– mod ). Given a nonpositivelygraded left DG-module N over Ω( X, End( E )) ∼ , one constructs the left CDG-module Q n N n over Ω( X, End( E )) with the action of Ω( X, End( E )) induced by the action ofΩ( X, End( E )) ∼ and the differential d = δ + d ′ . Remark.
For an affine variety X , the derived categories of filtered D X, E -moduleswith filtrations indexed by the integers (see Example 5.1) can be also described ascertain exotic derived categories of (the abelian categories of) graded modules overΩ( X, End( E )) ∼ . Namely, define the semiderived category (cf. [48]) of graded rightmodules over Ω( X, End( E )) ∼ (with the grading indexed by the integers) as the quo-tient category of the homotopy category of graded right modules over Ω( X, End( E )) ∼ by the thick subcategory formed by all complexes of graded modules that are coacyclicas complexes of graded modules over Ω( X, End( E )) ⊂ Ω( X, End( E )) ∼ . Then the de-rived category of the exact category of filtered right D X, E -modules with complete andcocomplete filtrations indexed by the integers (and compatible with the filtration F on D X, E ) is equivalent to the semiderived category of graded right modules overΩ( X, End( E )) ∼ . Analogously, define the semiderived category of graded left modulesover Ω( X, End( E )) ∼ as the quotient category of the homotopy category of gradedleft modules over Ω( X, End( E )) ∼ by the thick subcategory formed by all complexes hat are contraacyclic as complexes of graded modules over Ω( X, End( E )). Then thederived category of the exact category of filtered left D X, E -modules with completeand cocomplete filtrations indexed by the integers is equivalent to the semiderivedcategory of graded left modules over Ω( X, End( E )) ∼ . The functors providing theseequivalences of categories are defined as follows. The functors assigning complexesof graded modules over Ω( X, End( E )) ∼ to complexes of filtered D X, E -modules areconstructed exactly as in the nonnegative/nonpositive grading/filtration case above.When one constructs the functors assigning complexes of filtered modules over D X, E to complexes of graded modules over Ω( X, End( E )) ∼ , one has to pass to the comple-tion (for right modules) or the cocompletion (for left modules) with respect to thefiltration of D X, E -modules after performing the procedures described above for thenonnegative/nonpositive grading/filtration case. The functors of passing to the as-sociated graded gr F D X, E -modules on the derived categories of filtered D X, E -modulescorrespond to the forgetful functors assigning graded modules over Ω( X, End( E )) tograded modules over Ω( X, End( E )) ∼ (in both the left and right module situations).Here the derived category of graded gr F D X, E -modules is identified with the coderivedand contraderived categories of graded modules over Ω( X, End( E )). The analogousresults hold for arbitrary nonhomogeneous Koszul rings over base rings of finite ho-mological dimension (see [48]) in place of the filtered ring of differential operators.B.5. Coherent D -modules. Let X be a smooth algebraic variety over a field k and E be an algebraic vector bundle over X . Let D b ( mod coh – D X, E ) denote the boundedderived category of coherent (locally finitely generated) right D X, E -modules. Further-more, let D b ( mod cohfil – D X, E ) denote the bounded derived category of the exact categoryof filtered right D X, E -modules with locally finitely generated filtrations, i. e., coher-ent right D X, E -modules M with filtrations F by coherent O X -submodules compatiblewith the filtration F on D X, E and such that F n M = F n − M · F D X, E for large n .By [20, Exercise III.6.8] there are enough vector bundles on X , i. e., any coherentsheaf on X is the quotient sheaf of a locally free sheaf (of finite rank).Let D abs ( mod coh –Ω X ( E nd ( E ))) denote the absolute derived category of O X -coherentright CDG-modules over the quasi-coherent CDG-algebra Ω X ( E nd ( E )). Here a quasi-coherent right CDG-module N over Ω X ( E nd ( E )) is said to be O X -coherent if it hasa finite number of nonzero graded components only and all these graded componentsare coherent O X -modules. The absolute derived category D abs ( mod coh –Ω X ( E nd ( E )))is defined as the quotient category of the homotopy category of O X -coherent rightCDG-modules over Ω X ( E nd ( E )) by its minimal thick subcategory containing the totalCDG-modules of exact triples of O X -coherent right CDG-modules over Ω X ( E nd ( E )).Finally, let D ( mod cohgr –Ω X ( E nd ( E )) ∼ ) denote the derived category of O X -coherentsheaves of internally graded right DG-modules over Ω X ( E nd ( E )) ∼ , the O X -coherencebeing defined as above. lthough this is clearly not necessary for the next Theorem to be valid, it will beconvenient for us to presume our filtrations and internal gradings to be indexedby nonnegative integers, and incorporate this assumption into the definitions of D b ( mod cohfil – D X, E ) and D ( mod cohgr –Ω X ( E nd ( E )) ∼ ). Theorem. (a)
The triangulated categories D b ( mod coh – D X, E ) and D abs ( mod coh –Ω X ( E nd ( E ))) are naturally equivalent. (b) The triangulated categories D b ( mod cohfil – D X, E ) and D ( mod cohgr –Ω X ( E nd ( E )) ∼ ) arenaturally equivalent.Proof. There is a natural fully faithful functor D b ( mod coh – D X, E ) −→ D ( mod – D X, E ).The natural functor D abs ( mod coh –Ω X ( E nd ( E ))) −→ D co ( mod –Ω X ( E nd ( E ))) is alsofully faithful, as one can show in the way of the proof of Theorem 3.11.1. The naturalfunctors D b ( mod cohfil – D X, E ) −→ D ( mod ↑ fil – D X, E ) and D ( mod cohgr –Ω( X, E nd ( E )) ∼ ) −→ D ( mod ↑ –Ω( X, E nd ( E )) ∼ ) are also clearly fully faithful. We only have to show thatthe subcategories we are interested in correspond to each other under the equiva-lences of categories D ( mod – D X, E ) ≃ D co ( mod –Ω X ( E nd ( E ))) and D ( mod ↑ fil – D X, E ) ≃ D ( mod ↑ –Ω X ( E nd ( E )) ∼ ). It follows immediately from the constructions of the du-ality functors that they map D abs ( mod coh –Ω X ( E nd ( E ))) into D b ( mod coh – D X, E ) and D ( mod cohgr –Ω X ( E nd ( E )) ∼ ) into D b ( mod cohfil – D X, E ).Furthermore, for any vector bundle F over X the right D X, E -module F ⊗ O X D X, E corresponds to the trivial O X -coherent CDG-module F ⊗ O X E nd ( E ) overΩ( X, E nd ( E )). Analogously, for any n > D X, E -module F ( n ) ⊗ O X D X, E with the filtration F induced by the filtration F on D X, E and the filtration F on F ( n ) = F given by the rules F n − F ( n ) = 0, F n F ( n ) = F ( n ) correspondsto the O X -coherent sheaf of internally graded DG-modules F ( n ) ⊗ O X E nd ( E ) overΩ( X, E nd ( E )) ∼ living in the internal degree n and the cohomological degree 0. Soit remains to check that the filtered right D X, E -modules F ( n ) ⊗ O X D X, E generate D b ( mod cohfil – D X, E ), as it will then follow that the right D X, E -modules F ⊗ O X D X, E generate D b ( mod coh – D X, E ).It is important that the Rees algebra L j F j D X, E is Noetherian and has a finitehomological dimension. Given a coherent filtered right D X, E -module M , construct itsresolution · · · −→ Q −→ Q −→ M consisting of filtered D X, E -modules isomorphicto finite directs sums of filtered D X, E -modules of the form F ( n ) ⊗ O X D X, E . For d large enough, the embedding K −→ Q d − of the image of the morphism Q d −→Q d − will split locally in X as an embedding of filtered right D X, E -modules. Itfollows that F K · D X, E ⊂ K will be locally a direct summand of F Q d − · D X, E and K / ( F K · D X, E ) will be a direct summand of Q d − / ( F Q d − · D X, E ) as filtered right D X, E -modules, so one can prove that K is a direct sum of filtered D X, E -modules ofthe form F ( n ) ⊗ O X D X, E arguing by induction. nother approach is to notice that all four our “bounded-coherent” triangulatedcategories consist of compact objects in the corresponding larger triangulated cate-gories and generate them as triangulated categories with infinite direct sums. It fol-lows immediately by a result of [41] that the subcategories of compact objects in thelarger triangulated categories are obtained by adjoining to the “bounded-coherent”categories the images of all idempotent endomorphisms. So the duality functors iden-tify the desired “bounded-coherent” categories up to adjoining the images of idem-potents. Three of the four “bounded-coherent” categories are bounded derived cate-gories of Noetherian abelian categories, so they are closed under the images of idem-potents. To check that the absolute derived category of O X -coherent CDG-modules D abs ( mod coh –Ω( X, E nd ( E ))) also contains images of its idempotents, one has to use theforgetful functor D b ( mod cohfil – D X, E ) −→ D b ( mod coh – D X, E ) and the corresponding “to-talization” functor D ( mod cohgr –Ω( X, E nd ( E )) ∼ ) −→ D abs ( mod coh –Ω( X, E nd ( E ))). (cid:3) Remark.
All the results of this appendix are applicable to any filtered sheaf of rings( D , F ) over a separated Noetherian scheme X such that the associated graded sheaf ofrings gr F D is a quasi-coherent graded O X -algebra isomorphic to the tensor productof the algebra of endomorphisms of a vector bundle E and the symmetric algebraof a vector bundle L . To such a filtered algebra one can assign a quasi-coherentCDG-algebra structure on the tensor product of the algebra of endomorphisms of E and the exterior algebra of L ∗ , construct the duality functors, and then repeatverbatim the arguments above. In particular, this applies to any sheaf of twisteddifferential operators [4, section 2], the enveloping algebra corresponding to a centralextension of a Lie algebroid (cf. Example 6.6), etc. In fact, the only conditions onehas to impose on the associated graded algebra to ( D , F ) are the coherence andlocal freeness of the components, finite homological dimension, Noetherianness, andKoszulity. Even more general results can be found in [48, sections 0.4 and 11.8]. References [1] A. L. Agore. Limits of coalgebras, bialgebras and Hopf algebras. Electronic preprint arXiv:1003.0318 [math.QA] .[2] H. Bass. Finitistic dimension and a homological generalization of semi-primary rings.
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