Rank functions on triangulated categories
aa r X i v : . [ m a t h . R A ] J a n RANK FUNCTIONS ON TRIANGULATED CATEGORIES
J. CHUANG AND A. LAZAREV
Abstract.
We introduce the notion of a rank function on a triangulated category C which gen-eralizes the Sylvester rank function in the case when C = Perf ( A ) is the perfect derived categoryof a ring A . We show that rank functions are closely related to functors into simple triangulatedcategories and classify Verdier quotients into simple triangulated categories in terms of partic-ular rank functions called localizing . If C = Perf ( A ) as above, localizing rank functions alsoclassify finite homological epimorphisms from A into differential graded skew-fields or, moregenerally, differential graded Artining rings. To establish these results, we develop the theoryof derived localization of differential graded algebras at thick subcategories of their perfectderived categories. This is a far-reaching generalization of Cohn’s matrix localization of ringsand has independent interest. Contents
1. Introduction 11.1. Notation and conventions 32. Rank functions on triangulated categories 42.1. Periodic rank functions 62.2. Stability conditions and rank functions 82.3. Simple triangulated categories 93. Rank functions on perfect derived categories of ordinary rings 113.1. Sylvester rank functions 113.2. Derived rank functions 124. Further properties of rank functions 145. Derived localization of differential graded algebras 175.1. Derived localization of A -algebras 195.2. Homotopy coherence 205.3. Module localization 215.4. Derived localization of ordinary rings 225.5. Commutative rings 235.6. Small example 245.7. Group completion 245.8. Rank functions for derived localization algebras 255.9. Loops on p-completions of topological spaces 256. Localizing rank functions and fraction fields 256.1. Localizing rank functions for hereditary rings 266.2. Derived fields of fractions 28References 281. Introduction
The dimension of a vector space V over a (possibly skew-)field K is a basic characteristic of V and it is an elementary fact that it is an invariant, i.e. does not depend on the choice of abasis in V . However, further generalizations with K replaced with a noncommutative ring A , Key words and phrases.
Rank function, triangulated category, derived localization, differential graded algebra.This work was partially supported by EPSRC grants EP/T029455/1 and EP/T030771/1. nd V replaced with an A -module, are not straightforward. Indeed, there are examples of ringssuch that their free modules do not have a well-defined dimension, or rank. One possibilityto obviate this difficulty is to start with a homomorphism f : A → K where K is a skew-field(which allows one to associate to an A -module a K -module by tensoring up) and define the rankof a A -module as the rank of the corresponding K -module. Different homomorphisms f giverise to possibly different ranks. This suggests that ranks are closely related to homomorphismsinto skew-fields. This was made precise by Cohn and Schofield [Coh95, Sch85] by showing thatmaps f as above are in one-to-one correspondence with certain functions, called Sylvester rankfunctions, on finitely presented A -modules, defined intrinsically.A crucial part of the Cohn-Schofield theory is the method of matrix localization . Given aring A and a matrix M with entries in A , there exists another ring A [ M − ] supplied with amap A → A [ M − ] such that the matrix M becomes invertible over A [ M − ]; moreover A [ M − ]is universal in the sense that any other ring having this property factorizes uniquely through A [ M − ]. If M is a 1 × s ∈ A , then A [ M − ] reduces to A [ s − ], the usuallocalization of A at s . Furthermore, if A is a commutative ring, then general matrix localizationreduces to inverting a single element, namely the determinant of M ; however in the generalnoncommutative case no such reduction is possible.Our original motivation was to rework the Cohn-Schofield theory in a way intrinsic to thederived category of A . To this end, we formulate the notion of a rank function on an arbitrarytriangulated category C . Compared to the Sylvester rank function, our definition is simplerand, arguably, more natural. In the case C = Perf ( A ) for a ring A , it subsumes that of theSylvester rank function but does not reduce to it . The ‘exotic’ rank functions on Perf ( A ) (i.e.those that are not equivalent to Sylvester rank functions) are related to maps from A into graded skew-fields or graded simple Artinian rings in the homotopy category of differential graded (dg)rings . Recall that maps in the homotopy category of dg rings are computed by replacing thesource with a cofibrant dg ring. This needs to be done even if the source is an ordinary ring.In other words, such maps are invisible on the classical level.Our notion of a rank function on C appears close, albeit certainly not equivalent to, thenotion of cohomological functions on C in the sense of Krause [Kra16]. The precise relationshipbetween the two notions is unclear at the moment and we hope to return to this question infuture.Apart from the purely algebraic motivation described above, Sylvester rank functions are ofgreat relevance to geometric group theory and, in particular, to various versions of the Atiyahconjecture, cf. [JZ19] for a survey of recent results in this direction. A natural question, thatwe also leave for future investigation, is whether our notion of a rank functions can be usefullyexploited in that context.One unexpected source of rank functions on triangulated categories turns out to be Bridge-land’s ‘stability conditions’ [Bri07]. Namely, it turns out that there is a continuous map fromthe space of stability conditions to that of rank functions on the same triangulated category.Here we limit ourselves with merely recording this observation but it undoubtedly deservesfurther study.Next, we need to develop an analogue of matrix localization in a homotopy invariant way.Even when one is interested in inverting only one element in a ring, i.e. a 1 × A with respect to an arbitrary thick subcategory of Perf ( A ). This extends the notion of matrixlocalization since the latter is the nonderived version of the localization with respect to the thicksubcategory generated by a collection of free complexes of length 2. In contrast with invertinga single element, even for commutative rings general derived localization may have nontrivialderived terms. However, for hereditary algebras, derived localization reduces to the non-derivednotion. rank function ρ on a triangulated category C has a kernel Ker( ρ ), the thick subcategoryof C consisting of objects of rank zero. We describe those rank functions for which the Verdierquotient C / Ker( ρ ) is simple i.e. equivalent to the perfect derived category of a graded skew-field. These are the so-called localizing rank functions. It turns out that localizing rank functionsclassify homotopy classes of homological epimorphisms from dg rings into dg fields and dg simpleArtinian rings. We call such homological epimorphisms derived fraction fields . Strikingly,ordinary rings (even finite dimensional algebras over fields) have nontrivial derived fractionfields.There are, of course, exact functors from a given triangulated category C to a simple onethat are not Verdier quotients; these induce (non-localizing) rank functions on C . It is an openquestion to which extent such functors are captured by rank functions.The paper is organized as follows. In Section 2 the notion of a rank function on a triangulatedcategory is introduced, in two equivalent ways: as a function on objects or on morphismssatisfying appropriate conditions, as well as its refinement, a d -periodic rank function where d = 1 , , . . . , ∞ (an ordinary rank function is then 1-periodic). We show how rank functionscan be constructed using functors into simple triangulated categories and (very briefly) stabilityconditions.In Section 3 we prove that ∞ -periodic rank functions on the perfect derived category ofordinary rings are nothing other than Sylvester rank function whereas Section 4 establishesvarious further properties of rank functions reminiscent of those for Sylvester rank functions.Section 5 develops the theory of derived localization for dg algebras at thick subcategoriesof their perfect derived categories and compares it with the non-derived notion. This sectionhas an independent interest and can be read independently of the rest of the paper. FinallySection 6 introduces the notion of a localizing rank function and shows that they describe Verdierquotients into simple triangulated categories and homotopy classes of homological epimorphismsfrom dg algebras into dg fields and dg simple Artinian rings.1.1. Notation and conventions.
Throughout this paper we work with homologically gradedchain complexes over a fixed unital commutative ring k . Unadorned tensor products and Homswill be assumed to be taken over k . For a chain complex A we denote by Σ A its suspensiongiven by (Σ A ) i = A i − . The signs ≃ and ∼ = will stand for quasi-isomorphisms and isomorphismsof chain complexes respectively.We will normally use the abbreviation ‘dg’ for ‘differential graded’. By ‘dg algebra’ we willmean ‘dg associative unital algebra’ over k . The category of dg algebras dgAlg has the structureof a closed model category (with weak equivalences being multiplicative quasi-isomorphisms)and so it makes sense to talk about homotopy classes of maps between dg algebras. A given dgalgebra B together with a dg algebra map A → B (not necessarily central) will be referred toas an A -algebra. The category A ↓ dgAlg of A -algebras is likewise a closed model category. Thecategory of right dg A -modules over a dg algebra A will be denoted by Mod − A and we will referto its objects as A -modules. The category Mod − A is a closed model category whose homotopycategory is denoted by D ( A ), the derived category of A . We chose to focus (for notationalconvenience) on right modules but will occasionally use left modules as well, making sure thatno confusion would arise.For A -modules M, N we denote by Hom A ( M, N ) the chain complex of A -linear homomor-phisms M → N . We write RHom A ( M, N ) for the corresponding derived functor obtainedby replacing M with a cofibrant A -module quasi-isomorphic to M . If M = N we will writeEnd A ( M ) and REnd A ( M ) for Hom A ( M, N ) and RHom A ( M, N ) respectively.For a (right) A -module M and a left A -module N we write M ⊗ A N and M ⊗ LA N for theirtensor product and derived tensor product respectively. For two dg algebras B and C suppliedwith a dg algebra maps A → B and A → C their free product or pushout will be denoted by B ∗ A C , and its derived version – by B ∗ LA C . description of the closed model categories of algebras and modules convenient for ourpurposes is given in [BCL18], except for the minor difference that left modules are treated inthat paper whereas right modules are emphasized here.We will freely use the language of triangulated categories and their localizations, cf. [Kra10]for an overview. If C is a triangulated category with translation functor Σ and Σ d ∼ = id we saythat C has period d ; if no such d exists C is said to have infinite period. Given a triangulatedcategory C , its full triangulated subcategory S is thick if it is closed with respect to takingretracts. In this situation one can form the Verdier quotient C /S , a triangulated categorysupplied with a (triangulated) functor j : C → C /S whose kernel is S and universal with respectto this property. A triangulated subcategory S of C is called localizing if it contains all smallcoproducts; in that case the Verdier quotient often admits a right adjoint i : C /S → C , and thenthe endofunctor L := i ◦ j : C → C is called the (Bousfield) localization functor with respect to S . It is necessarily idempotent: L ∼ = L and for any X ∈ C the natural map X → L ( X ) is calledthe localization of X (with respect to S ). For a collection of objects S of C we will denote byLoc( S ) the smallest localizing subcategory of C containing S ; we say that S generates Loc( S ).We say that an object X is a (classical) generator of the triangulated category C if C is thesmallest thick subcategory of C containing X coincides with C .For a triangulated category C a perfect (or compact) object X is characterized by the propertyHom C ( X, ⊕ i ∈ I X i ) ∼ = ⊕ i ∈ I Hom C ( X, X i )for any collection of X i , i ∈ I of A -modules indexed by a set I . The full subcategory of D ( A )consisting of perfect A -modules will be denoted by Perf ( A ); note that A is a generator of Perf ( A ).A dg category is understood to be a category enriched over dg k -modules. Thus, for twoobjects X and X in a dg category C we have a dg space of homomorphisms Hom( X , X ) andcomposition is a dg map. The homotopy category H ( C ) of the dg category C has the sameobjects as C and for two objects X , X in C we have Hom H ( C ) ( X , X ) := H [Hom C ( X , X )].A dg functor F : C → C ′ between two dg categories is quasi-essentially surjective if H ( F ) : H ( C ) → H ( C ′ ) is essentially surjective and quasi-fully faithful if F induces quasi-isomorphismson the Hom-spaces; if both conditions are satisfied then F is called a quasi-equivalence .2. Rank functions on triangulated categories
Let C be a triangulated category. Definition 2.1. A rank function on C is an assignment to each object X of C of a nonnegativereal number ρ ( X ), such that the following conditions hold. Translation invariance: for any object X , we have(O1) ρ (Σ X ) = ρ ( X ); Additivity: for any objects X and Y , we have(O2) ρ ( X ⊕ Y ) = ρ ( X ) + ρ ( Y ); Triangle inequality: for any exact triangle X → Y → Z , we have(O3) ρ ( Y ) ≤ ρ ( X ) + ρ ( Z ) Remark 2.2.
Conditions (O1) and (O3) may be replaced by the statement that for any exacttriangle X → Y → Z , the triple ( ρ ( X ) , ρ ( Y ) , ρ ( Z )) is triangular, i.e. it is is composed ofthe side lengths of a (possibly degenerate) planar triangle. Indeed, one only needs to show thatthe latter condition together with (O2) implies (O1). Consider the exact triangle X −→ X → X ⊕ Σ X . Then the triangularity condition together with (O2) implies that ρ (Σ X ⊕ X ) = ρ (Σ X ) + ρ ( X ) ≤ ρ ( X )so that ρ (Σ X ) ≤ ρ ( X ). Similarly the exact triangle Σ X ⊕ X → Σ X −→ Σ X implies that ρ ( X ) ≤ ρ (Σ X ) and it follows that ρ ( X ) = ρ (Σ X ). ank functions on triangulated categories may alternatively be defined as functions on mor-phisms, as follows. Definition 2.3.
A rank function on C is an assignment to each morphism f in C of a nonnegativereal number ρ ( f ), such that the following conditions hold. Translation invariance: for any morphism f , we have(M1) ρ (Σ f ) = ρ ( f ); Additivity: for any morphisms f and g , we have(M2) ρ ( f ⊕ g ) = ρ ( f ) + ρ ( g ); Rank-nullity condition: for any exact triangle X f −→ Y g −→ Z , we have(M3) ρ ( f ) + ρ ( g ) = ρ (id Y );The translation between the two definitions is given by the following formulae:(2.1) ρ ( X ) = ρ (id X )(2.2) ρ ( f : X → Y ) = ρ ( X ) + ρ ( Y ) − ρ (cone( f ))2 . Proposition 2.4.
Definitions 2.3 and 2.1 are equivalent.Proof.
Given a nonnegative function ρ on morphisms in C satisfying the three conditions ofDefinition 2.3, the rule (2.1) defines a function on objects which is clearly nonnegative, andtranslation invariance (O1) and additivity (O2) follow immediately from the correspondingproperties (M1) and (M2). Finally, take an exact triangle X f −→ Y g −→ Z in C , with rotatedexact triangles Y g −→ Z h −→ Σ X and Σ − Z Σ − h −−−→ X f −→ Y . We have: ρ ( X ) + ρ ( Z ) − ρ ( Y ) = ρ (id X ) + ρ (id Z ) − ρ (id Y )= (cid:16) ρ (Σ − h ) + ρ ( f ) (cid:17) + ( ρ ( g ) + ρ ( h )) − ( ρ ( f ) + ρ ( g ))= 2 ρ ( h ) ≥ , by (M1) and (M3), which establishes (O3).Conversely, given a nonnegative function ρ on objects of C satisfying the three conditions ofDefinition 2.1, define a function on morphisms by formula (2.2). For f : X → Y , we have anexact triangle Y → cone( f ) → Σ X , so (O1) and (O3) imply that ρ ( f ) ≥
0. Properties (M1)and (M2) are consequences of (O1) and (O2), respectively, and, finally, given an exact triangle X f −→ Y g −→ Z , we confirm property M3: ρ ( f ) + ρ ( g ) = ρ ( X ) + ρ ( Y ) − ρ ( Z )2 + ρ ( Y ) + ρ ( Z ) − ρ (Σ X )2= ρ ( Y )= ρ (id Y ) , using O1. (cid:3) In view of Proposition 2.4, we regard a rank function on C as a function on both objects andmorphisms, related by equations (2.1) and (2.2).We call a rank function ρ • object-faithful , if for all nonzero objects X we have ρ (id X ) = 0. • morphism-faithful , if for all nonzero maps f we have ρ ( f ) = 0; • integral , if ρ ( f ) ∈ Z for all f . prime if ρ is integral and C admits a generator X such that ρ (id X ) = 1. Remark 2.5.
In the case C is a tensor triangulated category (i.e it has a symmetric monoidalstructure compatible with its triangulation, cf. [HPS97, Appendix A]), it makes sense to re-quire additionally that a rank function ρ is multiplicative in the sense that ρ (id X ⊗ id Y ) = ρ (id X ) ρ (id Y ) for any two objects X, Y ∈ C . Multiplicative rank functions are likely to be ofrelevance to tensor triangular geometry , [Bal05] but will not be considered in this paper.2.1.
Periodic rank functions.
We will consider a certain refinement of the notion of a rankfunction defined above.Let R be a commutative ring and fix d ∈ { , , . . . } ∪ {∞} . Put R ( d ) := ( R [ q, q − ] , if d = ∞ R [ q ] / ( q d − , if d < ∞ and R ≥ ( d ) := ( R ≥ [ q, q − ] , if d = ∞ R ≥ [ q ] / ( q d − , if d < ∞ For φ, ψ ∈ R ( d ), we write φ ≥ ψ to mean φ − ψ ∈ R ≥ ( d ). Given two integers q, q ′ where d is divisible by d ′ or d = ∞ , there is an obvious reduction map π d,d ′ : R ( d ) → R ( d ′ ). We willmostly be interested in the case R = R or R = Z . Definition 2.6. A d -periodic rank function on a triangulated category C is an assignment toeach morphism f in C of ρ ( f ) ∈ R ≥ ( d ), such that axioms (M2) and (M3) hold, axiom (M1)gets modified as follows: Translation invariance: for any morphism f , we have(Mp1) ρ (Σ f ) = qρ ( f );and, in addition, the following axioms hold: Triangular inequality: for all morphisms f , g and h in C such that g and h share thesame domain and f and h share the same codomain, we have(M4) ρ (cid:18) f h g (cid:19) ≥ ρ ( f ) + ρ ( g ); Ideal condition: for any morphisms f and g for which the composition gf is defined,we have(M5) ρ ( gf ) ≤ ρ ( f ) and ρ ( gf ) ≤ ρ ( g ) . Remark 2.7.
Note that given two integers d and d ′ where d ′ divides d or d = ∞ , a d -periodicrank function on C determines a d ′ -periodic rank function on C via the reduction map R ( d ) → R ( d ′ ). In particular, any d -periodic rank function gives rise to a 1-periodic rank function. Underthis reduction, axiom (Mp1) becomes (M1) and we will see that often (e.g. when d = 1) theaxioms (M4) and (M5) are consequence of axioms (M1), (M2) and (M3). In particular, theterms ‘rank function’ and ‘1-periodic rank function’ are synonymous. Furthermore, often aperiodic rank function can be defined as a function on objects (e.g. we saw that it holds for d = 1).A d -periodic rank function taking values in Z ≥ ( d ) will be called integral . The notions ofa prime, object-faithful and morphism-faithful rank function obviously make sense in the d -periodic case. Proposition 2.8.
Let ρ be a d -periodic rank function on C . For all objects X in C , put ρ ( X ) := ρ (id X ) ∈ R ≥ ( d ) . Then ρ satisfies the additivity axiom (O2) of the rank function whereas axioms (O1), (O3) getrefined as follows: Translation invariance: for any object X in C , we have (Op1) ρ (Σ X ) = qρ ( X ); riangle inequality: for any exact triangle X → Y → Z in C we have (Op3) ρ ( X ) − ρ ( Y ) + ρ ( Z ) = ( q + 1) φ, for some φ ∈ R ≥ ( d ) . More precisely, we can take φ = ρ ( f ) , where f : Σ − Z f −→ X isthe connecting morphism for the triangle.Moreover, if ρ is integral, i.e. ρ ( f ) ∈ Z ≥ ( d ) for all morphisms f in C , then ρ ( X ) ∈ Z ≥ ( d ) forany X ∈ C .Proof. This is a straightforward modification of the first part of proof of Proposition 2.4. Inparticular conditions (Op1) and (O2) follow immediately from (Mp1) and (M2). Furthermore,given an exact triangle Σ − Z f −→ X g −→ Y h −→ Z , we have, using (M3) and (Mp1), ρ ( X ) − ρ ( Y ) + ρ ( Z ) = ( ρ ( f ) + ρ ( g )) − ( ρ ( g ) + ρ ( h )) + ( ρ ( h ) + ρ (Σ f ))= ( q + 1) ρ ( f ) , as desired. The claim about integrality is likewise clear. (cid:3) Remark 2.9.
Note that in order to obtain axioms (Op1), (O2) and (Op3), we only make useof conditions (Mp1), (M2) and (M3) of Definition 2.6.It is natural to ask whether a rank function is determined by its values on objects (so as toobtain a periodic analogue Proposition 2.4). The following result gives a partial answer to that.
Proposition 2.10.
Assume that d = ∞ or d is odd and there is given an assignment to anyobject X of C of an element ρ ( X ) ∈ R ( d ) satisfying conditions (Op1), (O2) and (Op3). Then,for any f : X → Y in C the formula (2.3) ρ ( f ) := ρ ( Y ) − ρ (cone( f )) + qρ ( X ) q + 1 determines a d -periodic rank function on C . If, in addition, ρ ( X ) ∈ Z ( d ) and the polynomial φ figuring in triangle inequality (Op3), belongs to Z ( d ) , then the obtained rank function is integral.Proof. We will only treat the case of real rank functions; the integral case is obtained completelyanalogously. Note that q + 1 is invertible in R ( d ) if d is odd, and a non-zero-divisor (like allnon-zero elements) if d = ∞ . By Proposition 2.8, the element ρ ( Y ) − ρ (cone( f )) + qρ ( X ) isdivisible by q + 1 and it follows that it is uniquely divisible. Thus, ρ ( X ) ∈ R ( d ) for any X ∈ C .The conditions (Mp1), (M2) and (M3) of Definition 2.6 are proved by the same argument asthe second part of Proposition 2.4.To check (M5), suppose we are given morphisms f : X → Y and g : Y → Z . We have ρ ( f ) − ρ ( gf ) = qρ ( X ) + ρ ( Y ) − ρ (cone( f )) q + 1 − qρ ( X ) + ρ ( Z ) − ρ (cone( gf )) q + 1= ρ ( Y ) − ρ (cone( f ) ⊕ Z ) + ρ (cone( gf ))) q + 1 ≥ , where the inequality is deduced from Lemma 2.11 below and (Op3).Finally, we check (M4). Suppose we have morphisms f : X → Y , g : Z → W and h : Z → Y in C . There is an exact triangle in C of the formcone( g ) → cone (cid:18) f h g (cid:19) → cone( f ) . ence, by (O2) and (Op3), ρ (cid:18) f h g (cid:19) = ρ ( Y ⊕ W ) − ρ (cid:18) cone (cid:18) f h g (cid:19)(cid:19) + qρ ( X ⊕ Z ) q + 1 ≥ ρ ( Y ) + ρ ( W ) − ρ (cone( f )) − ρ (cone( g )) + qρ ( X ) + qρ ( Z ) q + 1= ρ ( f ) + ρ ( g ) . (cid:3) Lemma 2.11.
For any pair of composable morphisms f : X → Y and g : Y → Z in atriangulated category, there exists a exact triangle of the form Y → cone( f ) ⊕ Z → cone( gf ) . Proof.
Let X f −→ Y h −→ cone( f ) be an exact triangle containing the morphism f . The result is obtained by applying the octa-hedral axiom to the composition of the morphism ( h, g ) : Y → cone( f ) ⊕ Z and the projectioncone( f ) ⊕ Z → cone( f ). (cid:3) Remark 2.12.
An explicit form of the morphism cone( f ) ⊕ Z → cone( gf ) appearing in thestatement of Lemma 2.11 could be obtained under a strengthening of the axioms of a triangu-lated category; see [May01].2.2. Stability conditions and rank functions.
Recall that a stability condition on a tri-angulated category C , cf. [Bri07] is a pair ( P , Z ) where P = P ( φ ) , φ ∈ R is a slicing on C , acollection of subcategories of C with properties modelled on the Postnikov truncations in thecategory of chain complexes and a central charge , that is a homomorphism Z : K ( C ) → C compatible with the slicing in a suitable way. Every nonzero object E of C has a filtration0 = E → . . . → E n = E so that A i := cone( E i − → E i ) belongs to P ( φ i ) and φ > . . . > φ n ;one sets φ −P ( E ) = φ and φ + P ( E ) = φ n . The mass of E is defined as m σ ( E ) = P ni =1 | Z ( A i ) | .More generally, one can introduce a parameter and define m σ,t ( E ) = X | Z ( A i ) | e φ i t cf. [DHKK14, Section 4.5]. Proposition 2.13.
Given a stability condition σ = ( P , Z ) on a triangulated category C , themass m σ defines an object faithful rank function on C .Proof. Translation invariance (O1), additivity (O2) and object-faithfulness are obvious. Thetriangle inequality for m σ,t ( E ) is proved in [Ike16, Proposition 3.3] for all t and (O3) follows bytaking t = 0. (cid:3) Remark 2.14.
The last result suggests that a rank function may serve as a replacement fora stability condition on C which exists even in the absence of a t -structure (e.g. when C isperiodic).The set Stab( C ) of stability conditions on C is a topological space. The topology may beinduced by the generalized metric (allowed to assume infinite values): d ( σ , σ ) = sup = E ∈C (cid:26) | φ − σ ( E ) − φ − σ ( E ) | , | φ + σ ( E ) − φ + σ ( E ) | , (cid:12)(cid:12)(cid:12)(cid:12) log m σ ( E ) m σ ( E )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) , (cf. [Bri07], p. 341). In a similar (albeit much more obvious) way, the set FRank( C ) of objectfaithful rank functions on C is topologized by the generalized metric d ( ρ , ρ ) = sup = E ∈C (cid:12)(cid:12)(cid:12)(cid:12) log ρ ( E ) ρ ( E ) (cid:12)(cid:12)(cid:12)(cid:12) . n the set Rank( C ) of all rank functions on C , it is more useful, following Schofield [Sch85,Chapter 7], to consider the topology of pointwise convergence of real-valued functions on objectsof C .The following result is immediate: Proposition 2.15.
The maps
Stab( C ) → FRank( C ) ֒ → Rank( C ) σ m σ are continuous. (cid:3) In [BDL20], it is suggested that a compactification of the quotient Stab( C ) / C of Stab( C ) bya natural action of C may be constructed as the closure of its image in Rank( C ) / R × .2.3. Simple triangulated categories.
The most basic examples of rank functions come from simple triangulated categories.
Definition 2.16.
We say that a triangulated category C is simple if every exact triangle in C is split, and C is generated as a triangulated category by one non-zero indecomposable object.Simple triangulated categories are in 1-1 correspondence with graded fields , i.e. graded ringswhose non-zero homogeneous elements are invertible. Proposition 2.17.
A triangulated category C is simple if and only if C is equivalent to Perf ( K ) where K is a graded field.Proof. If C is a simple triangulated category with an indecomposable generator X then thegraded ring End ∗ ( X ) is a graded field. Conversely, all monomorphisms and epimorphisms ofgraded modules over a graded field K split and it follows that Perf ( K ) is a simple triangulatedcategory. (cid:3) Remark 2.18.
Graded fields can easily be classified in terms of ordinary skew-fields. Indeed,let K be a graded field; then K , its zeroth component is an (ungraded) skew-field. Supposethat K = K and let d be the smallest positive integer such that K has a nonzero, henceinvertible, element of degree d ; denote this element by t . Then K isomorphic as a K -moduleto K [ t, t − ] and if t is central then this isomorphism is multiplicative. In general K will beisomorphic to a skew Laurent polynomial ring K [ t, t − ] σ where σ is an automorphism of K and the multiplication is determined by the rule ta = σ ( a ) t for a ∈ K .The graded field of the form K = K [ t, t − ] σ with | t | = d will be called d -periodic and d is called the period of K ; furthermore we adopt the convention that an ungraded skew-field K = K has infinite period. The corresponding simple triangulated category Perf ( K ) is d -periodic i.e. the functor Σ d on Perf ( K ) is naturally isomorphic to the identity.Periodic rank functions on simple triangulated categories are essentially unique. More pre-cisely, we have the following result. Proposition 2.19.
The space of d ′ -periodic rank functions on a d -periodic simple triangulatedcategory is isomorphic to R ≥ ( d ′ ) if d = ∞ or d ′ is a divisor of d ; otherwise it is { } . Ifthe space is nonzero, there exists a unique (up to multiplication by a power of q ) prime rankfunction.Proof. Let C be a simple d -periodic triangulated category with a generator X and ρ be a non-zero d ′ -periodic rank function on C ; assume that d ′ < ∞ . Then ρ ( X ) ∈ R ≥ ( d ′ ) so that d ′ isthe smallest integer with q d ′ = 1. If d < ∞ then X ∼ = Σ d X and ρ ( X ) = ρ (Σ d X ) = q d ρ ( X ) sothat q d = 1 from which it follows that d ′ divides d . Furthermore, any other d ′ -periodic rankfunction can be obtained from ρ by multiplying it with an arbitrary element of R ≥ ( d ′ ) andrank functions corresponding to different elements in R ≥ ( d ′ ) will be different. The argumentwith d ′ = ∞ is similar. Finally, if a non-zero rank function on C exists, then the condition ρ ( X ) = 1 specifies it uniquely. (cid:3) orollary 2.20. A simple triangulated category admits a unique prime rank function.Proof.
This follows from Proposition 2.19 by specializing q = 1. (cid:3) Proposition 2.21.
If a triangulated category admits a morphism-faithful prime rank functionthen it is simple.Proof.
Let C be a triangulated category having a generator X and a morphism-faithful rankfunction ρ with ρ ( X ) = 1. Let f : Σ n X → X be any non-zero morphism of some degree n ;by (M3), ρ ( f ) is either 1 or 0, by morphism-faithfulness the second possibility is ruled out so ρ ( f ) = 1. Considering the exact triangleΣ − Y h / / Σ d X f / / X g / / Y and using (M3) again we conclude that ρ ( h ) = ρ ( g ) = 0 and then by morphism-faithfulness g = h = 0 and f is an isomorphism. Thus, any nonzero element in End( X ) is invertible, i.e.End( X ) is a graded skew-field. (cid:3) Definition 2.22.
A dg algebra A is called a dg skew-field if its homology H ( A ) is a gradedskew-field. Similarly A is a simple Artinian dg algebra if H ( A ) is a graded simple Artinianalgebra (i.e. it is isomorphic to the graded matrix algebra M n ( K ) over some graded skew-field K ).The derived categories of dg skew-fields and dg Artinian simple algebras are all simple: Proposition 2.23. (1)
Let A be a simple Artinian dg algebra so that H ( A ) ∼ = M n ( K ) where K is a gradedskew-field of period d and n is some integer. Then Perf ( A ) is a simple triangulatedcategory of period d . (2) Conversely, if A is a dg algebra for which Perf ( A ) is simple, then A is a simple dgArtinian algebra.Proof. The homology graded ring H ( A ) of A is a graded matrix algebra over some gradedskew-field K . A primitive idempotent e ∈ H ( A ) determines a retract X := eA in Perf ( A ). The H ( A )-module H ( X ) is a simple generator of the category of H ( A )-module. Given a perfectdg A -module M , its homology H ( M ) is a graded H ( A )-module that is a finite direct sum of H ( A )-modules H ( X ) and it is clear that this decomposition lifts to Perf ( A ) so that M is adirect sum of copies of X in Perf ( A ). Similarly, a map M → N of A -modules is determinedby the map H ( M ) → H ( N ) of H ( A )-modules and it follows that any exact triangle in Perf ( A )splits. Thus, Perf ( A ) is simple and its period clearly coincides with that of K . This proves (1).For (2) let A be a dg algebra for which Perf ( A ) is simple; denote by X an indecomposablegenerator of Perf ( A ) and assume without loss of generality that X is a cofibrant A -module.Since every nonzero map X → Σ n X is invertible in Perf ( A ), the dg algebra B := End A ( X ) is adg skew-field. Then the functor F : M M ⊗ A X is a dg equivalence between the categories ofcofibrant left A -modules and cofibrant B -modules and therefore the dg algebras A ∼ = End A ( A )and End B ( F ( A )) = End B ( X ) are quasi-isomorphic. Since B is a dg skew-field, the B -module X is a direct sum of simple B -modules and thus, End B ( X ) is a matrix algebra of a dg skew-field,i.e a dg simple Artinian ring. (cid:3) Remark 2.24.
A simple Artinian dg algebra or even a dg skew-field of finite period is not determined up to quasi-isomorphism by its homology algebra. Examples of non-formal A ∞ algebras whose homology algebras are skew-fields are not hard to construct (cf. for exampleeven Moore algebras of [Laz03]). This situation arises also in stable homotopy theory whereEilenberg-MacLane spectra of graded fields and Morava K -theories have equivalent simple ho-motopy categories of modules yet are very different in many ways, e.g. they have inequivalenthomotopy categories of bimodules. . Rank functions on perfect derived categories of ordinary rings
We start with a short review of ordinary Sylvester rank functions, following [Sch85]. Thistheory and its applications motivated us to develop its derived analogue.3.1.
Sylvester rank functions.
Let A be a ring. Denote by fp( A ) the category of finitely pre-sented A -modules, and by Proj ( A ) the subcategory of finitely generated projective A -modules. Definition 3.1. A Sylvester morphism rank function on A associates to each morphism f of Proj ( A ) a rank ρ ( f ) ∈ R ≥ . It is required to satisfy the following conditions. Normalization: (m1) ρ (id A ) = 1; Additivity: for any morphisms f and g , we have(m2) ρ (cid:18) f g (cid:19) = ρ ( f ) + ρ ( g ); Triangular inequality: for all morphisms f , g and h such that g and h share the samedomain and f and h share the same codomain, we have(m3) ρ (cid:18) f h g (cid:19) ≥ ρ ( f ) + ρ ( g ); Ideal condition: for any morphisms f and g for which the composition gf is defined,we have(m4) ρ ( gf ) ≤ ρ ( f ) and ρ ( gf ) ≤ ρ ( g ) Definition 3.2. A Sylvester module rank function on A associates to each finitely presented A -module M a rank ρ ( M ) ∈ R ≥ , such that Normalization: (o1) ρ ( A ) = 1; Additivity: for all finitely presented modules M and N , we have(o2) ρ ( M ⊕ N ) = ρ ( M ) + ρ ( N ); Triangle Inequality: for any exact sequence L → M → N → A -modules, we have(o3) ρ ( N ) ≤ ρ ( M ) ≤ ρ ( L ) + ρ ( N ) . Let P f −→ Q → M → , be an exact sequence of A -modules, with P, Q ∈ Proj ( A ), so that M ∈ fp( A ). The formulas ρ ( f ) = ρ ( Q ) − ρ ( M ) and ρ ( M ) = ρ (id Q ) − ρ ( f ) , yield a 1-1 correspondence between Sylvester morphism rank functions on A and Sylvesterobject rank functions on A . So we just call the resulting function defined on both morphismsand objects a Sylvester rank function on A . Remark 3.3.
A Sylvester rank function on A may be recovered from its values on homomor-phisms f : A m → A n between free modules, since every finitely presented module is isomorphicto the cokernel of such a map. By additivity, one may even restrict to the case of squarematrices, i.e. the case m = n . et S be a simple Artinian ring. Then S is isomorphic to a matrix algebra M n ( K ) over askew-field K . Every object of fp( S ) = Proj ( S ) is a finite direct sum of copies of the simplemodule V = K n , and the unique Sylvester rank function on fp( S ) takes the value n on V .Given a homomorphism A → B , we obtain a right exact functor fp( A ) → fp( B ) : M B ⊗ A M . It is easy to confirm that any Sylvester rank function on B pulls back via thisfunctor to a Sylvester rank function on A . In particular any homomorphism A → S into asimple Artinian ring determines a canonical Sylvester rank function on A . Two homomorphisms A → S and A → S ′ into simple Artinian rings are considered equivalent if they become equalafter composing with homomorphisms S → S ′′ and S ′ → S ′′ into a third simple Artinian ring.Equivalent homomorphisms clearly determine the same Sylvester rank function on fp( A ). Theorem 3.4.
There is a 1-1 correspondence between equivalence classes of homomorphismsof A into simple Artinian rings S and Q -valued Sylvester rank functions on A taking the value or on A/rA for any integer r . In this correspondence S is a skew-field if and only if thecorresponding rank function is Z -valued.Proof. See [Sch85], Chapter 7, Theorems 7.12 and 7.14. (cid:3)
Remark 3.5.
Let ρ be a Sylvester rank function on A . Since 0 ≤ ρ ( A/rA ) ≤ r , the condition on ρ ( A/rA ) is automatically satisfied if ρ is Z -valued. The same is true foran arbitrary Sylvester rank function on A when A is an algebra over a field, for then A/rA isisomorphic to either A or 0. See [Sch85], p. 121 for a discussion on this issue.We already explained how to obtain a rank function from a homomorphism into a simpleArtinian ring S , and since S is an algebra over a field, namely its centre, it will satisfy theadditional condition, c.f. Remark 3.5. The reverse construction in the case of an integralSylvester rank function ρ is described as follows. The Cohn localization of A ρ of A with respectto all morphisms f : P → Q in Proj ( A ) such that ρ ( f ) = ρ ( P ) = ρ ( Q ), can be shown to belocal with residue skew-field, say, K , cf. [Sch85, Theorem 7.5]. The desired homomorphism isthe composition A → A ρ → K .3.2. Derived rank functions.
Let A be an ordinary ring. We consider periodic rank func-tions on the triangulated category Perf ( A ). A d -periodic rank function ρ on Perf ( A ) is called normalized if ρ ( A ) = 1. We will denote by NRk d ( Perf ( A )) and Syl( A ) the set of normalized d -periodic rank functions on Perf ( A ) and Sylvester rank functions on fp( A ) respectively. In thissection we give a comparison between NRk d ( Perf ( A )) and Syl( A ). Theorem 3.6.
Let A be ordinary ring and d ∈ { , , . . . } ∪ {∞} . The restriction of anynormalized d -periodic rank function on Perf ( A ) to Proj ( A ) is a Sylvester morphism rank functionon A . In the case d = ∞ , this determines a 1-1 correspondence between normalized ∞ -periodicrank functions on Perf ( A ) and Sylvester rank functions on A .Proof. The first statement follows from a straightforward unraveling of definitions. The strongerstatement for d = ∞ is a consequence of the Proposition 3.8 below. (cid:3) Lemma 3.7.
Let ρ be a normalized d -periodic rank function on Perf ( A ) . Let X be a boundedcomplex of A -modules with finitely generated projective terms X n . Then ρ ( X ) ≤ X n ∈ Z ρ ( X n ) q n . Proof.
This follows by induction on the length of X from the triangle inequality (Op3). (cid:3) Proposition 3.8.
Let A be an ordinary ring. Any normalized ∞ -periodic rank function ρ on Perf ( A ) is determined by its restriction to Proj ( A ) . More precisely, given a bounded complex X = { X n } of finitely generated projective A -modules with a differential d n : X n → X n − , wehave (3.1) ρ ( X ) = X n ∈ Z ( ρ ( X n ) − ρ ( d n ) − ρ ( d n +1 )) q n . onversely, any Sylvester rank function on A extends (uniquely) via this formula to a rankfunction ρ on Perf ( A ) . This extension will be referred to as the derived rank function on Perf ( A ) .Proof. Let us prove that given a Sylvester rank function ρ , formula (3.1) gives a ∞ -periodicrank function on Perf ( A ). Note that given a complex X = { . . . d n ← X n d n +1 ← . . . } of finitelygenerated projective A -modules that is exact at X n , we have that ρ ( X n ) = ρ ( d n ) + ρ ( d n +1 ). Itfollows, firstly, that (3.1) defines the rank of a possibly unbounded complex that has vanishinghomology for all but finitely many degrees and, secondly, that the rank of an acyclic complexis zero.Given a quasi-isomorphism f : X → Y of bounded complexes of finitely generated projective A -modules we can factor it as X i → Z p → Y where i is a cofibration and p is a fibration ofcomplexes, (and both are quasi-isomorphisms). Note that Z may be unbounded but, beingquasi-isomorphic to a bounded complex, it has a well-defined rank ρ ( Z ). (It could also bechosen to be bounded as the mapping cylinder of f ). Since X and Y are deformation retractsof Y , it follows that ρ ( X ) = ρ ( Z ) = ρ ( Y ). Therefore, ρ is well-defined on Perf ( A ).It is clear that the function ρ on Perf ( A ) given by (3.1) satisfies the axioms (Op1) and (O2)of the rank function. To prove (Op3), consider an exact triangle X → Y → Z in Perf ( A ).Without loss of generality, we can assume that the map X → Y is a cofibration of complexesof A -modules, in other words, it is a degree-wise split injection, and Z ∼ = X/Y is a boundedcomplex of finitely generated projective A -modules. By the property (o2) of a Sylvester rankfunction we have for all n ∈ Z :(3.2) ρ ( Y n ) = ρ ( X n ) + ρ ( Z n ) . Denote by d Xn , d Yn , d Zn , n ∈ Z the differentials in the complexes X, Y and Z respectively. Thenby (m3) we have for all n ∈ Z :(3.3) ρ ( d Yn ) ≥ ρ ( d Xn ) + ρ ( d Zn )From (3.2) we deduce:(3.4) ρ ( X ) − ρ ( Y ) + ρ ( Z ) = X n ∈ Z ( ρ ( d Yn ) + ρ ( d Yn +1 ) − ρ ( d Xn ) − ρ ( d Xn +1 ) − ρ ( d Zn ) − ρ ( d Zn +1 )) q n . By (3.3) the polynomial ρ ( X ) − ρ ( Y )+ ρ ( Z ) has positive coefficients; moreover it clearly vanishesat q = −
1. It follows that ρ ( X ) − ρ ( Y ) + ρ ( Z ) = ( q + 1) φ with φ ∈ R ≥ ( d ) as required.We will now prove that, conversely, any ∞ -periodic rank function ρ on Perf ( A ) is necessarilygiven by the formula (3.1). Given an interval I of integers, denote by X I ∈ Perf ( A ) the brutaltruncation of X concentrated in degrees n ∈ I , i.e. ( X I ) n = X n for n ∈ I , and ( X I ) n = 0otherwise, with differential d n equal to that of X for all n such that { n, n − } ∈ I . Sincethe claimed formula for ρ ( X ) is consistent with condition (Op1), it suffices to prove that theconstant term of ρ ( X ) is equal to ρ ( X ) − ρ ( d ) − ρ ( d ) . From the standard exact triangleΣ − X [2 , ∞ ) φ → X ( −∞ , → X → X [2 , ∞ ) we obtain by (Op3):(3.5) ρ ( X ( −∞ , ) − ρ ( X ) + ρ ( X [2 , ∞ ) ) = ( q + 1) ρ ( φ ) . Next note that by Lemma 3.7, the constant terms of ρ ( X [2 , ∞ ) and of Σ − X [2 , ∞ ) are zero andthen the constant term of ρ ( φ ) is likewise zero since ρ ( φ ) ≤ ρ (Σ − X [2 , ∞ ) ) by axiom (M3). Itfollows from equation (3.5) that the constant terms of ρ ( X ) and ρ ( X ( −∞ , ) coincide.Arguing similarly with the exact triangle → X ( −∞ , → X [ − , → Σ X ( −∞ , − , we find that the constant terms of ρ ( X [ − , ) and X ( −∞ , coincide and thus, also coincide withthe constant term of ρ ( X ). rite ρ ( X [ − , ) = aq − + b + cq , ρ ( X [ − , ) = a ′ q − + b ′ , and ρ ( X [0 , ) = b ′′ + c ′′ q . From thetwo exact triangles X [ − , → X [ − , → Σ X and Σ − X − → X [ − , → X [0 , , and the triangle inequality (Op3) we find that a = a ′ , c = c ′′ and ρ ( X [ − , ) q = − − ρ (Σ X − ) q = − + ρ ( X [0 , ) q = − = 0. Furthermore, from the exact triangle X → X [0 , → Σ X we obtain ρ ( X [0 , ) q =0 = ρ ( X ) − ρ ( d ) and similarly qρ ( X [ − , ) q =0 = ρ ( X − ) − ρ ( d ). We thencalculate b = ρ ( X [ − , ) q = − + a + c = ρ (Σ X − ) q = − + ρ ( X [0 , ) q = − + a ′ + c ′′ = − ρ ( X − ) + b ′′ + a ′ = ρ ( X [0 , ) q =0 + ( qρ ( X [ − , )) q =0 − ρ ( X − )= ρ ( X ) − ρ ( d ) + ρ ( X − ) − ρ ( d ) − ρ ( X − )= ρ ( X ) − ρ ( d ) − ρ ( d )as desired. (cid:3) Remark 3.9. • Let A be an ordinary ring. Let d < ∞ . To any normalized ∞ -periodic rank function ρ on Perf ( A ) we can assign the normalized d -periodic rank function π ∞ ,d ◦ ρ on Perf ( A ). Thisis a injective map with a canonical splitting: given a normalized d -periodic rank function ρ , restrict it to a Sylvester rank function on A , and then take the unique extension to anormalized ( ∞ -periodic) rank function ˆ ρ on Perf ( A ) provided by Proposition 3.8.NRk ∞ ( Perf ( A )) / / ∼ ' ' ❖❖❖❖❖❖❖❖❖❖❖ NRk d ( Perf ( A )) w w ♦♦♦♦♦♦♦♦♦♦♦ Syl( A ) • Suppose β : A → K is a map in the homotopy category of dg rings from a ring A to a d -periodic skew-field K . Let ρ = ρ β be the normalized d -periodic rank function on Perf ( A )obtained via pullback along β of the unique normalized d -periodic rank function on theperfect derived category of K . Then ˆ ρ may be described as follows. The homomorphism H ( β ) : A → K obtained by passing to homology algebras factors as the compositionof a homomorphism β : A → K and the inclusion K ֒ → K , and we have ˆ ρ = ρ β ,the pullback along β of the unique normalized ( ∞ -periodic) rank function on Perf ( K ).Moreover ρ = π ∞ ,d ◦ ˆ ρ if and only if (the homotopy class of maps) β is realized as anactual ring homomorphism A → K (and then, it must necessarily be H ( β )).4. Further properties of rank functions
As before, we assume that C is a triangulated category with a d -periodic rank function ρ .We will examine the behavior of ρ with respect to functors in or out of C . Given anothertriangulated category C ′ and an exact functor F : C ′ → C , the pullback ρ ′ = F ∗ ( ρ ), assigning toeach morphism f of C ′ the rank ρ ′ ( f ) = ρ ( F ( f )), is clearly also a d -periodic rank function on C ′ . The pullback of an integral rank function is integral.On the other hand, a pushforward of ρ is not always possible; later on we will consider thecase of a pushforward along a Verdier quotient.The factorizations f ◦ id X and id Y ◦ f of a morphism f : X → Y yield inequalities ρ ( f ) ≤ ρ ( X )and ρ ( f ) ≤ ρ ( Y ), by (M5). This motivates the following definitions. Definition 4.1.
Let ρ be a d -periodic rank function on C . We say that a morphism f : X → Y in C is left ρ -full if ρ ( f ) = ρ ( X ); • right ρ -full if ρ ( f ) = ρ ( Y ); • ρ -full if it is left full and right full. Lemma 4.2.
Let ρ be a d -periodic rank function on C . (1) Let X f −→ Y g −→ Z be an exact triangle in C , with a connecting homomorphism h : Σ − Z → X . Then • f if left ρ -full if and only if g is right ρ -full if and only if ρ ( h ) = 0 ; • f is ρ -full if and only if ρ ( Z ) = 0 . (2) Let f : X → Y and g : Y → Z be morphisms in C . If f is ρ -full, then ρ ( gf ) = ρ ( g ) . If g is ρ -full, then ρ ( gf ) = ρ ( f ) .Proof. The first part follows directly from the definitions. For the second part, consider theexact triangle cone( f ) → cone( gf ) → cone( g ) , given by the octahedral axiom. Suppose that f is ρ -full. Then ρ (cone( f )) = 0 and therefore ρ (cone( gf )) = ρ (cone( g )). We deduce that( q + 1)( ρ ( gf ) − ρ ( g )) = ( ρ ( X ) − ρ (cone( gf )) + qρ ( Z )) − ( ρ ( Y ) − ρ (cone( g )) + qρ ( Z )) = 0 . Since, by (M5), ρ ( gf ) − ρ ( f ) ≥
0, we deduce that ρ ( gf ) − ρ ( g ) = 0, as desired. The argumentwhen g is ρ -full is similar. (cid:3) Corollary 4.3.
Let α be an invertible element in k and f : X → Y is a morphism in C . Then ρ ( α · f ) = ρ ( f ) .Proof. The multiplication by α is an automorphism of X and so, it is a ρ -full map. Then theconclusion follows from Lemma 4.2 (2). (cid:3) Corollary 4.4.
The full subcategory on
Ker( ρ ) := { X ∈ C : ρ ( X ) = 0 } is a thick subcategory of C . Moreover, ρ descends to a d -periodic rank function ρ on the Verdierquotient C / U by any thick subcategory U ⊆ ker( ρ ) , and the pullback of ρ to C is ρ . The obtained d -periodic rank function on C / ker( ρ ) is object-faithful.Proof. The category C / U is the localization of C at a set of ρ -full morphisms, and so anymorphism f : M → N in C/ U can be written as a roof M h / / L N g o o where h and g aremorphisms in C and cone( g ) is an object of U (so, in particular, g is ρ -full). We set ρ ( f ) := ρ ( h );then part (2) of Lemma 4.2 ensures that ρ is a well-defined function on morphisms of C / U .Clearly, ρ pulls back to ρ , and satisfies (Mp1) and (M2). That ρ obeys (M3)-(M5) follows fromthe fact that every triangle or diagram of the form X → Y → Z or X → Y ← Z → W in C / U is isomorphic to the image of a triangle or diagram of the same form in C .The object-faithfulness of ρ on C / ker( ρ ) is clear. (cid:3) Remark 4.5.
Let d and d ′ be positive integers with d divisible by d ′ ; clearly the only elementin R ( d ) mapping to zero under the reduction map R ≥ ( d ) → R ≥ ( d ′ ) is zero. Thus, givena d -periodic rank function ρ and the corresponding d ′ -periodic rank function ρ ′ , their kernelscoincide. In particular, if one is interested in the thick subcategories that are kernels of rankfunctions, it results in no loss of generality to consider only ordinary (i.e. 1-periodic) rankfunctions. Lemma 4.6.
Let f and g be morphisms in C with the same domain and codomain. Then ρ ( f + g ) ≤ ρ ( f ) + ρ ( g ) . roof. We have a factorization ( f + g ) = (cid:0) (cid:1) (cid:18) f g (cid:19) (cid:18) (cid:19) . The desired inequality follows from axioms (M5) and (M2). (cid:3)
Corollary 4.7.
For any pair of objects
X, Y of C , define Hom ρ C ( X, Y ) = { f ∈ Hom C ( X, Y ) | ρ ( f ) = 0 } . This is an ideal of morphisms in C , by (M5) and by Lemma 4.6. (cid:3) Lemma 4.8.
Let f and g be morphisms of C with ρ ( g ) = 0 . Then ρ ( f + g ) = ρ ( f ) .Proof. By Lemma 4.6 we have ρ ( f + g ) ≤ ρ ( f ) + ρ ( g ) = ρ ( f ). Taking into account that ρ ( − g ) = ρ ( g ) by Corollary 4.3 we have ρ ( f ) = ρ ( f + g − g ) ≤ ρ ( f + g ) + ρ ( g ) = ρ ( f + g ) whichimplies the desired equality. (cid:3) Lemma 4.9.
Let ρ be a prime rank function on a triangulated category C supplied with agenerator X and denote by L ρ ( X ) the image of X in the Verdier quotient C / ker ρ . Then thegraded algebra End • ( L ρ ( X )) is local.Proof. Recall that the induced rank function ρ on C / ker ρ object-faithful. The set I := f ∈ End( L ρ ( X )) : ρ ( f ) = 0 is an ideal in End • ( L ρ ( X )) by (M5). If f ∈ End • ( L ρ ( X )) is not in I then ρ ( f ) = 1, in other words f : L ρ ( X ) → L ρ ( X ) is a ρ -full map. By Lemma 4.2 the cone of f has zero rank and so is zero in C / ker ρ ; thus f is invertible. Since any element in End • ( L ρ ( X ))not in I is invertible, it follows that I is a maximal ideal and End • ( L ρ ( X )) is local. (cid:3) So any prime rank function on a triangulated category with a generator X gives rise to amap End • ( X ) → F ρ , where F ρ is the (graded, skew) residue field of End • ( L ρ ( X )).Recall that the idempotent completion of an additive category C is a category ˜ C whose objectsare pairs ( X, e ) where X is an object of C and e is an idempotent endomorphism of X ; amorphism between two such pairs ( X, e ) and (
Y, t ) is a morphism f : X → Y in C such that ef = f = f t . The idempotent completion of a triangulated category is known to be triangulatedand it possesses a universal property with respect to exact functors into idempotent completetriangulated categories [BS01]. Lemma 4.10.
Any d -periodic rank function ρ : C → R ( d ) on a triangulated category C extendsuniquely to a d -periodic rank function on its idempotent completion.Proof. Given a map f : ( X, e ) → ( Y, t ) in ˜ C we define its rank as the rank of f : X → Y in C .Note that under this definition the rank of the identity morphism of ( X, e ) is ρ ( e ). All axiomsof the rank function except (M3) are obvious. To check (M3), consider an exact triangle in ˜ C :( X, e ) f → ( Y, t ) g → ( Z, k ) and note that it is a direct summand in ˜ C of the following exact triangle in C (4.1) ( X, f → ( Y, g ′ → ( Z ′ , Moreover, there is the following isomorphism in ˜ C :( Z ′ , ∼ = ( Z, k ) ⊕ (Σ X, − e ) ⊕ ( Y, t )and the morphism g ′ : ( Y, → ( Z ′ ,
1) can be represented as the composite map( Y, ∼ = / / ( Y, t ) ⊕ ( Y, − t ) g ⊕ id ( Y, − t ) / / ( Z, k ) ⊕ (Σ X, − e ) ⊕ ( Y, − t ) rom which it follows that ρ ( g ′ ) = ρ ( g ) + ρ (1 − t ). The exact triangle (4.1) gives ρ ( t ) + ρ (1 − t ) = ρ (( Y, t ) ⊕ ( Y, − t ))= ρ ( Y )= ρ ( f ) + ρ ( g ′ )= ρ ( f ) + ρ ( g ) + ρ (1 − t )and it follows that ρ ( t ) = ρ ( f ) + ρ ( g ) as desired. (cid:3) Assume now that C has a generator X and ρ is a prime rank function. An object of C which is a finite coproduct of shifted copies of X will be called graded free . Note that a mapbetween graded free objects could be written as a rectangular matrix M whose entries are gradedendomorphisms of X , so we can speak of the rank ρ ( M ) of M , generalizing the familiar notionin linear algebra. We will denote by M the matrix obtained from M by passing to the gradedresidue field F ρ of the graded local algebra End L ρ ( X ) and its usual graded rank by rank F ρ ( M ). Proposition 4.11.
We have ρ ( M ) = rank F ρ ( M ) .Proof. We may assume that ρ is object-faithful, so that End • ( X ) is graded local, otherwisereplace C with the Verdier quotient C / Ker( ρ ). By the usual row reduction process, M = ER ,where E is an invertible matrix and R is a rectangular matrix such that R is of the form (cid:18) I
00 0 (cid:19) , where I is an identity matrix. Here E is a product of permutation matrices and upperand lower triangular matrices with invertible elements on the diagonal. So it suffices to provethat ρ ( R ) = rank F ρ ( R ). This follows from Lemma 4.8. (cid:3) Corollary 4.12.
Let M be a matrix over End • ( X ) representing a morphism between gradedfree objects. The M has a (square) ρ -full submatrix N with ρ ( N ) = ρ ( M ) .Proof. This follows from the corresponding result for the graded ranks of matrices over a gradedskew-field. (cid:3) Derived localization of differential graded algebras
Let A be a dg algebra assumed, without loss of generality, to be cofibrant, and τ be a thicksubcategory of Perf ( A ). Definition 5.1.
A dg A -module M is called τ -local if RHom A ( X, M ) = 0 for any X ∈ τ . Lemma 5.2.
Let B be a dg algebra supplied with a map A → B . Then B is τ -local if and onlyif B ⊗ LA X ≃ for any X ∈ τ .Proof. Note that RHom A ( X, B ) ≃ RHom B ( X ⊗ LA B, B ), thus Hom B ( X ⊗ LA B, B ) ≃ A ( X, B ) ≃
0. Conversely, since X is a perfect dg A -module, X ⊗ LA B is a perfect dg B -module and so 0 ≃ RHom A ( X, B ) ≃ RHom B ( X ⊗ LA B, B ) implies that X ⊗ LA B ≃ (cid:3) Definition 5.3.
The derived localization of A with respect to τ is a dg algebra L τ ( A ) togetherwith a map A → L τ ( A ) making it a τ -local A -module and such that for any dg algebra B and a map f : A → B making B a τ -local A -module, there is a unique up to homotopymap L τ ( A ) → B making the following diagram commutative in the homotopy category of dgalgebras:(5.1) A f / / (cid:15) (cid:15) BL τ ( A ) < < ②②②②② emark 5.4. Let s ∈ A be an n -cycle of a dg algebra A for n ∈ Z , and let A/s be thehomotopy cofiber of the left multiplication map Σ n A → A . Then A/s is a perfect A -moduleand the localization with respect to the thick subcategory h A/s i generated by it, exists and isgiven explicitly by the formula L s A := L h A/s i ( A ) = A ∗ L k [ s ] k h s, s − i , cf. [BCL18]. We will nowgeneralize this result to an arbitrary thick subcategory. Theorem 5.5.
For any dg algebra A and any thick subcategory τ of Perf ( A ) , the derivedlocalization L τ ( A ) exists and is unique up to homotopy.Proof. Let us first suppose that τ is generated by a single perfect object X ∈ Mod − A , assumedwithout loss of generality to be cofibrant. Denote by E the dg algebra of endomorphisms of the A -module A ⊕ X . Let e be the element in E given by the projection E → A along X followed bythe inclusion of A into E . Then e is a zero-cocycle and an idempotent of E . We will show thatthe desired (derived) localization L τ ( A ) may be constructed as L e E , the (derived) localizationof E at e , cf. Remark 5.4. We have the following commutative diagram of dg categories: Mod − A F / / Mod − E { X } / / (cid:31) ? O O { E/e } (cid:31) ? O O (5.2)Here F ( M ) = Hom A ( A ⊕ X, M ), for M ∈ Mod − A , E/e is the cofiber of the left multiplication by e on E , { X } the full dg subcategory of Mod − A containing X and closed with respect to arbitrarycoproducts, homotopy cofibers and passing to quasi-isomorphic modules and similarly { E/e } is the full dg subcategory of Mod − A containing E/e , having arbitrary coproducts, homotopycofibers and closed with respect to passing to quasi-isomorphic modules. Note that
E/e isquasi-isomorphic as an E -module to (1 − e ) E ⊕ Σ(1 − e ) E and F ( X ) = Hom A ( A ⊕ X, X ) ∼ = X ⊕ Hom A ( X, X ) ∼ = (1 − e ) X. Therefore the image of { X } under F is { E/e } and the commutativity of 5.2 indeed holds.Since A ⊕ X is a compact generator of D ( A ), the functor F is a quasi-equivalence, as well asits restriction { X } → { E/e } .Let us construct (a homotopy class of) a map of dg algebras A → L e E . Note that thefunctor F could be viewed as tensoring over A with the left A -module Hom A ( A ⊕ X, A ) ∼ = eE .Composing F with the localization functor into Mod − L e E we obtain the functor G : Mod − A → Mod − L e E : M M ⊗ A ( eE ⊗ E L e E ) ∼ = M ⊗ A eL e E. Since G is a dg functor there is an induced (homotopy class of a) map of dg algebras f : A ∼ = End A ( A ) → End L e E ( eL e E ) ≃ End L e E ( L e E ) ∼ = L e E. The commutative diagram (5.2) implies X ⊗ A L e E ≃ G ( X ) ≃ F ( X ) ⊗ E L e E ≃ , meaning that L e E is h X i -local.Let us now prove that L e E has the required universal property with respect to maps from A into h X i -local algebras. To this end, let ˜ A := A × k . Then ˜ A is a dg algebra with the differential d ( a, x ) = d A ( a ) and the product ( a, x ) · ( a ′ , y ) = ( aa ′ , xy ) for a, a ′ ∈ A, x, y ∈ k . There is a dgalgebra map ˜ f : ˜ A → E given by the diagonal embedding ( a, x ) ( l a , − e ) where l a is theaction of A on itself by the left multiplication by a . Note that ˜ f could also be described as amap ˜ A ∼ = End ˜ A ( ˜ A ) → End L e E ( L e E ) ∼ = E induced by the functor − ⊗ ˜ A E : Mod − ˜ A → Mod − E .The idempotent (1 , ∈ ˜ A is mapped to e ∈ E under ˜ f , and, slightly abusing notation, wewill denote it also by e . We obtain an induced map on derived localizations L e ( ˜ A ) → L e ( E ).This produces a homotopy class of dg algebra maps A → L e ( E ) since clearly L e ( ˜ A ) ≃ A . This s the same map (up to homotopy) as f as implied by the following diagram of dg categoriesand dg functors, commutative up to a natural isomorphism. Mod − ˜ A −⊗ ˜ A L e E / / −⊗ ˜ A A (cid:15) (cid:15) Mod − L e E Mod − A −⊗ A eE / / Mod − E −⊗ E L e E O O Let B be a dg algebra supplied with a map g : A → B and h X i -local (so that X ⊗ A B ≃ g extends uniquely (up to homotopy) to a map L τ ( A ) := L e E → B .Consider the following diagram of dg algebras, commutative up to homotopy (excluding thedotted arrow).(5.3) ˜ A ˜ f / / (cid:15) (cid:15) E ∼ = End A ( A ⊕ X ) (cid:15) (cid:15) " " A ≃ L e ˜ A L e ( ˜ f ) / / g - - L e E = L h X i ( A ) * * ❯❯❯❯❯❯❯❯ B ≃ End B ( B ⊕ X ⊗ A B )Here the arrow from E to B is induced by the functor − ⊗ A B : Mod − A → Mod − B . By theuniversal property of the localization L e E the dotted arrow, making commutative the uppertriangle of (5.3), exists. It follows that the lower triangle of (5.3) commutes upon restriction to˜ A . However since the map ˜ A → A ≃ L e E becomes an isomorphism upon inverting e , and the dgalgebra B is e -inverting, we conclude that the lower triangle of (5.3) is commutative. Conversely,a similar argument shows that any dotted arrow, making the lower triangle commutative, makesthe upper trangle commutative and is, therefore, unique.So, the theorem is proved under the assumption that τ is generated by a single perfect object X . It is easy to see that for two perfect objects X, Y ∈ Mod − A , we have L h X ⊕ Y i ( A ) ≃ L h X i ( A ) ∗ LA L h Y i ( A )as both sides satisfy the required universal property with respect to any dg algebra B localwith respect to both X and Y . Letting X s , s ∈ S be a collection of compact generators of τ indexed by a set S , we can set A τ := ` LA,s ∈ S L h X S i ( A ), the (derived) coproduct over A ofderived localizations L h X S i ( A ). (cid:3) Derived localization of A -algebras. Recall that if C is a closed model category and X is an object of C then the undercategory of X is the category X ↓ C with objects Y ∈ C supplied with a map X → Y and morphisms being obvious commutative triangles in C . Theundercategory of X inherits the structure of a closed model category from C ; in the case when X is cofibrant, this undercategory is homotopy invariant in the sense that for any other cofibrant X ′ and a weak equivalence X → X ′ the undercategories X ↓ C and X ′ ↓ C are Quillen equivalent. Lemma 5.6.
For a τ -local A -algebra B there exists a map of A -algebras L τ ( A ) → B , uniqueup to homotopy in the undercategory A ↓ dgAlg .Proof. As in the proof of Theorem 5.5 we first consider the case when τ is generated by a singleperfect A -module X and arguing similarly with diagram (5.3), we conclude that there exists amap g : L τ → B making the diagram of dg algebras (which is a fragment of the diagram (5.3))(5.4) A " " ❊❊❊❊❊❊❊❊❊ (cid:15) (cid:15) L τ ( A ) g / / B omotopy commutative in the category of dg algebras. Thus, g could be viewed as a map inthe undercategory A ↓ dgAlg . The uniqueness of g is proved similarly: suppose that there existsanother map of A -algebras g ′ : L τ ( A ) → B making the (5.4) homotopy commutative. Then g and g ′ are homotopic in A ↓ dgAlg if and only if they are homotopic in ˜ A ↓ dgAlg and this, inturn, is equivalent to them being homotopic in E ↓ dgAlg . But they are indeed homotopic in E ↓ dgAlg by the defining property of the derived localization L e E , cf. [BCL18, Definition 3.3].The proof is finished as that of Theorem 5.5. (cid:3) Remark 5.7.
Another way to formulate Lemma 5.6 is to say that L τ ( A ) is the initial objectof the subcategory of the homotopy category of A ↓ dgAlg consisting of τ -local A -algebras. Itmay appear that this statement and its proof are just rephrasing the corresponding parts of thestatement and proof of Theorem 5.5. The substantive difference is that two maps of A -algebrasmay be homotopic as maps of dg algebras but not as maps in the undercategory A ↓ dgAlg . Wewill discuss this discrepancy in more detail below.Let now B be an A -algebra, assumed, without loss of generality, to be cofibrant in A ↓ dgAlg (meaning that the given map A → B is a cofibration of dg algebras). The induction functor − ⊗ A B : Mod − A Mod − B takes Perf ( A ) into Perf ( B ) and (abusing the notation slightly) wewill denote by the image of the thick subcategory τ ∈ Perf ( A ) under this functor, by the samesymbol τ . Proposition 5.8.
We have a natural isomorphism L τ ( B ) ∼ = B ∗ LA L τ ( A ) in the homotopycategory of A ↓ dgAlg .Proof. This is similar to [BCL18, Lemma 3.7]. There is a Quillen adjunction A ↓ dgAlg ⇄ B ↓ dgAlg with the left adjoint − ∗ A B and the right adjoint being the restriction functor. It is easyto see that this adjunction restricts to an adjunction between τ -local A -algebras and τ -local B -algebras and the corresponding homotopy categories. Since left adjoints preserve initial objects,the desired statement follows from Lemma 5.6. (cid:3) Corollary 5.9.
The map of dg algebras L τ ( A ) → L τ ( A ) ∗ LA L τ ( A ) given by the inclusion of theeither factor is a quasi-isomorphism. Recall from [Hov99, Chapter 5], for any closed model category C , the notion of a derivedmapping space , a simplicial set Map( X, Y ) where
X, Y ∈ C , generalizing the usual simplicialmapping space in a simplicial model category. Recall also from [Mur16] that a morphism f : X → Y in a model category is said to be a homotopy epimorphism if for any object Z , theinduced morphism Map( Y, Z ) → Map(
X, Z ) is an injection on connected components of thecorresponding simplicial sets and an isomorphism on homotopy groups for any choice of a basepoint. Then we have the following generalization of [BCL18, Proposition 3.17], which follows,as in op.cit. from Corollary 5.9.
Corollary 5.10.
The localization map A → L τ ( A ) is a homotopy epimorphism. (cid:3) Homotopy coherence.
We defined the derived localization L τ ( A ) of a dg algebra A through a certain universal property formulated in the homotopy category of dg algebras. Itmakes sense to ask whether more structured notions, taking into account the ∞ -structure ofthe category of dg algebras (i.e. the homotopy type of mapping spaces) can reasonably beconsidered. We will show that such, apparently more refined, versions of derived localization,nevertheless, turns out to be equivalent to the one defined above. Proposition 5.11.
Let C be an τ -local A -algebra. The following statements are equivalent: (1) C is isomorphic in the homotopy category of A ↓ dgAlg to L τ ( A )(2) For any τ -local A -algebra B there is a unique map C → B in the homotopy category of A ↓ dgAlg . (3) For any τ -local A -algebra B the map Map( L τ ( A ) , B ) → Map(
A, B ) induced by thelocalization map A → L τ ( A ) is a weak equivalence. For any τ -local A -algebra B the mapping space Map A ( L τ ( A ) , B ) in A ↓ dgAlg is con-tractible.Proof. There is the following homotopy fiber sequence of simplicial sets:(5.5) Map A ( L τ ( A ) , B ) → Map( L τ ( A ) , B ) → Map(
A, B );here Map A ( L τ ( A ) , B ) is the homotopy fiber over the map A → B that determines B as an A -algebra. The second map in (5.5) is a weak equivalence if and only if its homotopy fiber iscontractible over every point. Thus, (3) and (4) are equivalent. The implication (2) ⇒ (1) isimplied by the following fragment of the long exact sequence of the fibration (5.5): → π Map A ( L τ ( A ) , B ) → π Map( L τ ( A ) , B ) → π Map(
A, B ) , and the reverse implication (1) ⇒ (2) is Lemma 5.6. The implication (3) ⇒ (1) is obvious.Finally, the implication (1) ⇐ (3) follows from Corollary 5.10. (cid:3) Remark 5.12.
Let us call a map B → C in A ↓ dgAlg a τ -local equivalence if for any τ -local A -algebra X there is a weak equivalence Map A ( C, X ) → Map A ( B, X ). Then Proposition 5.11implies that for an A -algebra B its derived localization L τ ( B ) ≃ L τ ( A ) ∗ LA B is the Bousfieldlocalization of B in A ↓ dgAlg with respect to τ -local equivalences, cf. [Hir03] regarding thisnotion.5.3. Module localization.
We will now relate the notion of derived localization of algebrasto the Bousfied localization of D ( A ). Localization functors exist for a large class of triangulatedcategories C and thick subcategories S . For example, such is the case when C = D ( A ), thederived category of a dg algebra and S = Loc( τ ) where τ is a perfect thick subcategory of D ( A ).The localization of M ∈ Mod − A with respect to a thick subcategory τ ∈ Perf ( A ) is a τ -local A -module N together with a map f : M → N that is a local τ -equivalence, i.e. for any τ -local A -module L the induced map f ∗ : RHom( N, L ) → RHom(
M, L ) is a quasi-isomorphism. Alocalization of an A -module is clearly defined up to a quasi-isomorphism and, (slightly blurringthe distinstion between the category Mod − A and D ( A )) we will refer to it as the localization of M and denote by L Mod − Aτ ( M ). The following results connect module localization and (derived)algebra localization, generalizing the corresponding results in [BCL18]. Proposition 5.13 ( [Dwy06, Proposition 2.5], [BCL18, Theorem 4.12]) . There is a dg algebra X supplied with a dg algebra map A → X such that X ≃ L Mod − Aτ ( M ) as an A -module.Proof. Let τ ⊗ Perf ( A ⊗ A op ) (i.e. perfect A -bimodules) generatedby A -bimodules of the form X ⊗ A op with X ∈ τ . Then the argument of [BCL18, Theorem4.12] shows that L Mod − Aτ ( M ) is quasi-isomorphic as an A -module to REnd A op ( L Mod − A ⊗ A op τ ⊗ ( A ))and the latter is clearly an A -algebra. (cid:3) The following result is proved in [Dwy06, Proposition 2.10].
Proposition 5.14.
Let M be an A -module. Then M ≃ M ⊗ LA A → M ⊗ LA L Mod − Aτ A is thelocalization L Mod − Aτ ( M ) of M . (cid:3) Corollary 5.15.
The Quillen adjunction
Mod − A → Mod − L Mod − Aτ A with left adjoint givenby extension of scalars M M ⊗ LA L Mod − Aτ A and right adjoint given by restriction along A → L Mod − Aτ A induces an equivalence between D ( L Mod − Aτ A ) and the full subcategory of D ( A ) of τ -local modules.Proof. By Proposition 5.14 the functor M M ⊗ LA L Mod − Aτ A is the τ -localization of the A -module M ; thus if M is already τ -local, then M → M ⊗ LA L Mod − Aτ A is a quasi-isomorphism.Moreover, since L Mod − Aτ A , and L Mod − Aτ A is τ -local, any L Mod − Aτ A -module is also τ -local as lyingin in the localizing subcategory generated by any L Mod − Aτ A . Therefore, for any L Mod − Aτ A -module M the map M ⊗ LA L Mod − Aτ A → M is a quasi-isomorphism. (cid:3) heorem 5.16. If L Mod − Aτ ( A ) is a dg A -algebra which is the localization of A as an A-modulethen it is also the localization of A as a dg algebra (so that L Mod − Aτ ( A ) and L τ ( A ) are isomorphicin the homotopy category of A -algebras).Proof. This theorem is proved in [BCL18] in the special case when τ is generated by a set of A -modules having the form of a cofiber of an endomorhism of A , however the proof continues tohold for arbitrary τ . For the reader’s convenience we will repeat the main points. First, we provethat for any A -algebra C that is τ -local as an A -module there is a (homotopy class of a) map of A -algebras L Mod − Aτ ( A ) → C . This is Lemma 41.7 of [BCL18] and the proof applies verbatim. Sincethe algebra localization L τ ( A ) is τ -local, there is an A -algebra map f : L Mod − Aτ ( A ) → L τ ( A ).Next, by the universal property of L τ ( A ) and since L Mod − Aτ ( A ) is a τ -local A -algebra, there isa map g : L τ ( A ) → L Mod − Aτ ( A ). The composition f ◦ g : L τ ( A ) → L τ ( A ) is an endomorphismof L τ ( A ) as an A -algebra and should, therefore, be homotopic to the identity. Similarly thecomposition g ◦ f is an endomorhism of L Mod − Aτ ( A ) that is homotopic to the identity. Thus, f and g are mutually inverse quasi-isomorphisms of dg algebras L Mod − Aτ ( A ) and L τ ( A ). (cid:3) Definition 5.17.
A map of dg algebras, A → L τ ( A ) is called a finite homological epimorphism corresponding to a thick subcategory τ ∈ Perf ( A ). Remark 5.18.
A dg algebra map A → B is called a homological epimorphism , [Pau09] ifthe map B ⊗ LA B → B , induced by the multiplication on B , is a quasi-isomorphism. Clearlythe derived localization map A → L τ A is a homological epimorphism (since the map L τ ( A ) ⊗ LA L τ ( A ) → A is the τ -localization of L τ ( A ) but L τ ( A ) is already τ -local) but not every homologicalepimorphism is of this form, owing to the failure of the so-called telescope conjecture , cf. [Kel94]. Corollary 5.19.
Let A be a dg algebra. Then there is a 1-1 correspondence between: • thick subcategories in Perf ( A ) , • equivalence classes of homotopy classes of finite homological epimorphisms from A wheretwo such A → B and A → B ′ are equivalent if there is a homotopy commutative diagram A (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ ❆❆❆❆❆❆❆ B / / B ′ where B → B ′ is a quasi-isomorphism.Proof. Given a thick subcategory τ we construct a finite homological epimorphism A → L τ ( A ).Conversely, associate to a finite homological epimorphism A → B the kernel of the functor − ⊗ A B : Perf ( A ) → Perf ( B ). The universal property of the derived localization L τ ( A ) impliesthat these constructions are mutually inverse, as claimed. (cid:3) Derived localization of ordinary rings.
Let A be an ordinary algebra and τ be athick subcategory in Perf ( A ) generated by a collection of objects represented by complexes offinitely generated projective A -modules of length two . In other words, the localization map Perf ( A ) → Perf ( A ) /τ inverts some maps between finitely generated projective A -modules. Ifthese modules are free, such maps are represented by matrices with entries in A and derivedlocalization map A → L τ A is the derived version of Cohn’s matrix localization [Coh95]. Themore general (still underived) version belongs to Schofield [Sch85]. Definition 5.20.
Let S be a collection of maps between finitely generated projective A -modules.An algebra B supplied with a map A → B is called S -inverting if the functor ? B ⊗ A ? carriesevery morphism in S into an isomorphism of B -modules. The localization of A at S is the S -inverting algebra A [ S − ] such that for any other S -inverting S -algebra B the map A → B factors uniquely through A [ S − ].It is clear that A [ S − ] is unique if it exists. The existence of A [ S − ] is [Sch85, Theorem4.1]. The following result establishes a precise relationship between the derived and underivednotions (and simultaneously gives an independent proof of the existence of A [ S − ]). heorem 5.21. Let A , S and τ be as above. Then L τ A is a connective dg algebra i.e. H n ( L τ ( A )) = 0 for n < and H ( L τ ( A )) = A [ S − ] .Proof. By Theorem 5.16 we can identify L τ ( A ) with L Mod − Aτ ( A ). With this, the desired resultis the combination of [Proposition 3.1] [Dwy06] and [Proposition 3.2] [Dwy06] (note that op.cit.works with left modules but this difference is, of course, unimportant). (cid:3) It is natural to ask when Cohn-Schofield localization coincides with derived localization. Thefollowing result answers this question.
Proposition 5.22.
Let A , S and τ be as above. Then the canonical map L τ A → A [ S − ] is aquasi-isomorphism if an only if A [ S − ] is stably flat over A i.e. Tor An ( A [ S − ] , A [ S − ]) = 0 for n > .Proof. After the identification of L τ ( A ) and L Mod − Aτ ( A ), this is proved in [Proposition 3.3][Dwy06]. (cid:3) Corollary 5.23.
Let A be right-hereditary, i.e. having right global dimension ≤ . Thenthe derived localization L τ at any thick subcategory τ of Perf ( A ) is (quasi-isomorphic to) itsCohn-Schofield localization.Proof. This follows at once from Proposition 5.22 since higher Tor functors over a hereditaryalgebra vanish. (cid:3)
Commutative rings.
Assume that A is an ordinary commutative ring. Derived local-izations corresponding to two-term complexes A → A given by multiplications by elements of A are ordinary localizations of A as a commutative ring at a multiplicatively closed subset.However for more general thick subcategories τ , it is possible for L τ ( A ) to be a genuine dgalgebra. Recall that the well-known Hopkins-Neeman-Thomason theorem [Tho97] gives theclassification of all thick subcategories in Perf ( A ): these correspond bijectively to the unions ofclosed subsets in Spec( A ) having quasi-compact complement (such subsets are sometimes called Hochster open sets). Specifically, to any such subset S one associates the thick subcategory of Perf ( A ) consisting of perfect A -modules having support on S . Then one has the following result,in which we A stands for the structure sheaf on Spec A and we identify D ( A ) with the categoryof complexes of quasi-coherent sheaves of A -modules. Proposition 5.24.
Let τ be the thick subcategory corresponding to a Hochster open subset S of Spec A and denote by i : S ⊂ Spec( A ) the corresponding inclusion map. Then L τ ( A ) ∼ = Ri ∗ i ∗ ( A ) as objects of D ( A ) .Proof. Let D S ( A ) be the subcategory of D ( A ) consisting of complexes of quasi-coherent sheaveson Spec( A ) supported on S . It is clear that the (homotopy) fiber of the natural map A → Ri ∗ i ∗ A is supported on S . On the other hand, the sheaf Ri ∗ i ∗ A is τ -local; indeed for any A -modulesheaf F supported on S we have i ∗ F ≃ Ri ∗ i ∗ A , F ) ≃ RHom( i ∗ ( A ) , i ∗ F ) ≃ Ri ∗ i ∗ A is the localization of A with respect to the localizing subcategory D S ( A ). Now the desired statement follows from Theorem 5.16. (cid:3) Example 5.25.
Let A := k [ x, y ] be the polynomial algebra in two variables and τ be the thicksubcategory generated by the 1-dimensional A -module k [ x, y ] / ( x, y ). It is easy to see (e.g. usingthe Koszul complex) that L τ ( A ) is a dg algebra whose homology is concentrated in degrees 0and − A its derived localization L τ ( A ) isalso such. As usual, it is better to consider E ∞ algebras rather than strictly commutative ones.Then a positive answer to this question could be derived by combining the results of [EKMM97]and [Man03]. Since E ∞ algebras are rather tangential to the main themes of the present paper,we will only sketch the proof and omit all topological and operadic prerequisites, referring tothe two above mentioned sources for details. roposition 5.26. If A is a (dg) E ∞ algebra, then L τ ( A ) is also a dg E ∞ algebras and thelocalization A → L τ ( A ) can be constructed as a map of dg E ∞ algebras.Proof. Let H k be the Eilenberg-MacLane spectrum corresponding to the ring k ; it is known tobe a commutative S -algebra. According to [Man03, Theorem 7.11], there is a functor Ξ : B Ξ( B ) from the homotopy category of commutative H k algebras to the homotopy category of dg E ∞ algebras and another one R : M R ( M ) from the homotopy category of Ξ( B )-modules tothe homotopy category of B -modules. Moreover, both Ξ and R are equivalences. Additionally,the B -module R (Ξ( B )) is weakly equivalent to B for any S -algebra B ; this property is notstated explicitly in op.cit. but follows readily from the construction.It suffices to show that L Mod − Aτ is quasi-isomorphic to a dg E ∞ algebra and, by the setupdescribed above this is equivalent to showing that the Bousfield localization of (Ξ) − ( A ) is acommutative S -algebra (and the map into it from (Ξ) − ( A ) is that of commutative S -algebras).But this is proved in [EKMM97, Chapter 8, Theorem 2.2]. (cid:3) Small example.
The following is the smallest example of a finite-dimensional algebrapossessing nontrivial derived localization. Let A be the algebra with a basis e , e , α , α sothat e = e , e = e , α e = α = e α , e α = α = α e and the rest of the products are zero.The algebra A is the path algebra of a quiver with two vertices and two arrows between themrunning in the opposite directions, subject to the relations above. The elements α i , i = 1 , e i , i = 1 , B := REnd A ( k , k ) where A acts on k via e = α i = 0 , i = 1 ,
2. Also set B ′ :=REnd A ( k × k , k × k ) where A acts on k × k via α i = 0 , i = 1 ,
2. It is immediate that B ′ is(quasi-isomorphic to) the path algebra of the same quiver with arrows marked by the generators α ′ i , i = 1 , | α ′ i | = − B ≃ e B ′ e is the polynomial algebraon one generator β = α ′ α ′ with | β | = −
2. It further follows that REnd B ( k , k ) ≃ k [ α ] / ( α ),the exterior algebra on one generator α with | α | = 1.On the other hand, it is clear that L e A is the localization of A as an A -module with respect tothe functor RHom A ( − , e Ae ) ≃ RHom A ( − , k ) and the latter localization is (quasi-isomorphicto) REnd B ( k , k ), cf. [DG02, Theorem 2.1, Proposition 4.8] for this kind of statement.All told, we conclude that L e A ≃ k [ α ] / ( α ). The nonderived localization A [ e − ] of A is, ofcourse, the ground ring k .Next, consider the projective A modules e A and e A ; then the left multiplication with α determines a map e A → e A which we will regard as an object in Perf ( A ). Denote by τ thethick subcategory generated by this object. Then L τ A is isomorphic to M ( k ), the 2 × k and since M ( k ) is flat over A , we conclude that no higher derived terms arepresent, i.e. L τ ( A ) ≃ M ( k ). Similar conclusions can be made regarding derived localizationsof A at e and at the object e A → e A determined by the left multiplication by α .5.7. Group completion.
Let M be a discrete monoid and BM be its classifying space. Thenthe based loop space Ω BM is the so-called group completion of M and according to MacDuff’stheorem, any topological space X is weakly equivalent to some Ω BM [McD79]. The chainalgebra C ∗ (Ω BM ) is quasi-isomorphic to the derived localization of the monoid algebra k [ M ]at all monoid elements by [BCL18, Theorem 10.3]. It follows that the derived localization ofan ordinary (ungraded) algebra, such as k [ M ] can be a fairly arbitrary dg algebra. Here is aparticularly striking example due to Fiedorowicz [Fie84]. Example 5.27.
Let M be the monoid with five elements { , x ij , i, j = 1 , } which multiplyaccording to the rule x ij x kl = x il . It is easy to see that the non-derived group completionof M is trivial and that H ∗ ( M, k ) := Tor k [ M ] ∗ ( k , k ) coincides with the homology of S , thetwo-dimensional sphere. It follows that BM is weakly equivalent to S and, therefore Ω BM isweakly equivalent to Ω S . The homology of Ω S is k [ x ] with | x | = 1, and this dg algebra isclearly formal. Thus, the derived localization of k [ M ] is (quasi-isomorphic to) k [ x ]. Note thatthe localization map k [ M ] → C ∗ (Ω BM ) ≃ k [ x ] is highly nontrivial in the homotopy category f dg algebras (e.g. it induces a Verdier quotient on the level of derived categories) and it is not the one that factors through k .5.8. Rank functions for derived localization algebras.
Let A be a dg algebra and ρ be arank function on Perf ( A ). Consider the map A → L ρ ( A ) from A into its derived localization atKer( ρ ). Then the following result holds. Theorem 5.28.
The rank function ρ descends to an object-faithful rank function ρ on Perf ( L ρ ( A )) (so that the pullback of ρ under the direct image functor Perf ( A ) → Perf ( L ρ ( A )) is ρ ).Proof. By Corollary 4.4, ρ descends to an object-faithful rank function on the Verdier quotient Perf ( A ) / Ker( ρ ). Note that Perf ( L ρ ( A )) is the idempotent completion of Perf ( A ) / Ker( ρ ) by[Nee92, Theorem 2.1] and so the obtained rank function on Perf ( A ) / Ker( ρ ) extends further to ρ on Perf ( L ρ ( A )) by Lemma 4.10. Clearly, ρ has the required properties. (cid:3) Remark 5.29.
The above theorem is an extension of the corresponding statement for Sylvesterrank functions, [Sch85, Theorem 7.4]. This is a key result in theory of Sylvester rank functionsand its proof in op.cit. is very involved. The almost trivial proof of the much more general resultabove demonstrates the advantage of the notion of a rank function for triangulated categoriesover the classical notion.5.9.
Loops on p-completions of topological spaces.
Another example of derived localiza-tion in topology comes from the study of chain algebras of based loops on completed classifyingspaces of finite groups, cf. [CL98, Ben09]. Let X be a topological space such that π ( X ) isfinite (e.g. the classifying space of a finite group) and X ∧ p is the p -completion of X . Then it isproved in [CL20] that there is an idempotent e ∈ F p [ π ( X )] such that the derived localization L e F p [ π ( X )] is quasi-isomorphic, as a dg algebra, to C ∗ Ω( X ∧ p ), the chain algebra of the basedloop space of X ∧ p . Remark 5.30.
When X is the classifying space of a finite group, this result (in a somewhatdifferent formulation) was proved in the paper [Vog17] where a good portion of derived local-ization theory was also developed. Unfortunately, the present authors had not been aware ofthis earlier work and did not make a proper attribution to it in [BCL18, CL20].6. Localizing rank functions and fraction fields
We will start with the following almost obvious result.
Proposition 6.1.
Let f : X → Y be a morphism in a triangulated category C supplied with arank function ρ such that ρ ( f ) = 0 . Then the following conditions are equivalent: (1) The morphism f factors through an object of rank zero. (2) There exists an object Z in C and ρ -full morphism g : Z → X such that f ◦ g = 0 . (3) There exists and object W in C and a ρ -full morphism h : Y → W such that h ◦ f = 0 . (4) The morphism f maps to zero under the Verdier quotient map C → C / ker( ρ ) .Proof. The equivalence of (1) with (2) and (3) follows from Lemma 4.2 and the equivalence of(2) and (3) with (4) follows from the characterization of morphisms in a Verdier quotients interms of left or right fractions. (cid:3)
Definition 6.2.
An integral rank function on a triangulated category C is localizing if anymorphism in C of rank zero satisfies either of the equivalent conditions of Proposition 6.1 Remark 6.3.
The notion of a localizing rank function is motivated by constructing derivedlocalization of dg rings to dg simple Artinian rings. If one is interested in maps into dg algebrasmore general than dg simple Artinian rings (e.g. dg analogues of von Neumann regular rings),one can speculate that real-valued rank functions will be relevant. In this context perhaps it ismore natural to require that any rank 0 morphism factors through objects of arbitrarily smallrank. We will not elaborate on this more subtle notion in the present paper however. heorem 6.4. Let C be a triangulated category admitting a generator. Then there is a bijectionbetween the following two sets: • localizing prime rank functions; • thick subcategories of C with a simple Verdier quotient.Proof. Let τ be a thick subcategory of C such that C /τ is simple. Fix an indecomposable object X in C /τ . Then there is a unique morphism-faithful rank function on C /τ taking value 1 on X ,see Corollary 2.20. The pullback of this rank function to C is a localizing rank function on C .Conversely, given a localizing rank function ρ : C → Z , let ρ be the rank function on C / Ker( ρ )induced by ρ ; since ρ is localizing, ρ is morphism faithful and ρ ( X ) = 1 for some generator X .By Proposition 2.21, C / Ker( ρ ) is simple.It is clear that the two processes described define mutually inverse maps between the twosets in the statement of the theorem. (cid:3) The following result gives a complete description of homotopy classes of derived localizationsof dg algebras into dg skew-fields or, more generally, dg simple Artinian rings, in terms of rankfunctions. Classically, only partial results of this sort were available (e.g. for a very specific classof rings or a particular class of localizations), cf. [Sch85, Theorems 5.4, 5.5] and [Coh95, Theorem4.6.14].
Theorem 6.5.
Let A be a dg algebra. Then there is a bijection between the following two sets: • localizing prime rank functions on Perf ( A ) . • equivalence classes of homotopy classes of finite homological epimorphisms A → B intosimple Artinian dg algebras B .Moreover, ρ ( A ) = 1 if and only if B is a dg skew-field.Proof. By Theorem 6.4 localizing prime rank functions ρ on Perf ( A ) correspond bijectively tothick subcategories on Perf ( A ) with a simple Verdier quotient. For such a thick subcategory τ the image of A in Perf ( A ) is a generator M of Perf ( A ) /τ that we can assume without loss ofgenerality to be cofibrant over A . Since Perf ( L τ ( A )) is the idempotent completion of Perf ( A ) /τ and Perf ( A ) /τ is simple, Perf ( L τ ( A )) is also simple and it follows by Proposition 2.23 that L τ ( A ) is a simple Artinian dg algebra. By Corollary 5.19 such thick subcategories correspondbijectively with the equivalence classes of homotopy classes of finite homological epimorphisms A → L τ ( A ). Finally, the condition ρ ( A ) = 1 means that M is an indecomposable object of Perf ( A ) /τ and in this case H ( L τ ( A )) ∼ = End Perf ( A ) /τ ( M, M ) must be a graded skew-field. (cid:3)
Example 6.6.
Let A be the 4-dimensional algebra of §5.6. The localization map A → L τ ( A ) ≃ M ( k ) induces a functor Perf ( A ) → Perf ( M ( k )) where the target triangulated category issimple. The induced rank function on Perf ( A ) is localizing.On the other hand, consider the localization map A → L e ( A ) ≃ k [ α ] /α . The category Perf ( k [ α ] /α ) is not simple, but composing with the augmentation k [ α ] /α → k , we obtain amap A → k . Note that the latter map is the Cohn-Schofield (nonderived) localization of A .The induced rank function (which corresponds to a certain Sylvester rank function for A ) is not localizing.6.1. Localizing rank functions for hereditary rings.
Localizing rank functions are easiestto construct for perfect derived categories of (right)-hereditary algebras in which case theyessentially reduce to the nonderived notion.Let A be an ordinary k -algebra and K ( A ) be its Grothendieck group of its category offinitely generated projective modules. The abelian group K ( A ) has a pre-order specified bythe declaring the classes of finitely generated projective modules in K ( A ) to be positive. Thenwe have the notion of a projective rank function on A cf. [Sch85]. Definition 6.7.
A projective rank function on A is homomorphism of pre-ordered groups ρ : K ( A ) → R for which ρ [ A ] = 1. Sylvester rank function on A (which, by Theorem 3.6 is equivalent to a normalized ∞ -periodic rank function on Perf ( A )) restricts to a function on the positive cone of K ( A ) withvalues in nonnegative real numbers and thus, determines a projective rank function on A . Onecan ask whether, conversely, one can associate to a projective rank function on A a Sylvesterrank function on A . For this, a rank needs to be assigned to any map between two finitelygenerated projective A -modules. One can attempt the following definition. Definition 6.8.
Let A be a ring with a projective rank function ρ . Given a map f : P → Q between two finitely generated projective A -modules, its inner rank ρ ( f ) is defined as ρ ( f ) :=inf( ρ ( S )) where S ranges through finitely generated projective A modules through which f factors.There is no reason for an inner rank function to be Sylvester in general. However, this holdsin one important special case. Proposition 6.9.
Let A be a hereditary algebra with a projective rank function ρ . Then theassociated inner rank function is Sylvester.Proof. We prove the desired statement by quoting relevant results of [Sch85] (all of which areelementary and easy). Firstly, by [Sch85, Theorem 1.11] the Sylvester law of nullity holds for ρ ; that is if α : P → P and β : P → P are two maps between finitely generated projective A -modules for which β ◦ α = 0 then ρ ( α ) + ρ ( β ) ≤ ρ ( P ).Next, the law of nullity implies axioms (m2) and (m3) of the Sylvester morphism rankfunctions, by [Sch85, Lemmata 1.14, 1.15]. Finally, the axioms (m1) and (m4) are immediatefrom the definition. (cid:3) Remark 6.10.
As is clear from the above proof, Proposition 6.9 is essentially contained in[Sch85], although it is not explicitly formulated in op. cit. as such. Moreover, the statementof the proposition holds under the weaker assumption that A be weakly semihereditary (asopposed to hereditary). We will not need this stronger result. Corollary 6.11.
Let A be a hereditary algebra with a projective rank function ρ . Then ρ extendsuniquely to a rank function on Perf ( A ) .Proof. The inner rank function associated to ρ is Sylvester by Proposition 6.9. Next, Proposition3.8 implies that a Sylvester rank function gives rise to an ∞ -periodic rank function on Perf ( A )and the latter determines, by reduction, a (1-periodic) rank function on Perf ( A ). (cid:3) Remark 6.12.
For a hereditary algebra A there is a 1-1 correspondence between rank functionsand ∞ -periodic rank functions on Perf ( A ). Indeed, this is straightforward to check on complexesof finitely generated projective A -modules of length 2 using formula (3.1) and the hereditaryproperty implies that such complexes generate the whole triangulated category Perf ( A ). Proposition 6.13.
Let A be a hereditary algebra and ρ be a rank function on Perf ( A ) associated,as above, with a projective rank function on A . Then ρ is localizing.Proof. The rank function ρ is localizing if and only if the derived localization L ρ A of A atKer ρ is a dg skew-field by Theorem 6.5 and by Corollary 5.23 L ρ A is quasi-isomorphic to theunderived localization of A at the collection of ρ -full maps between finitely generated A -modules.By [Sch85, Theorem 5.4] this underived localization is a skew-field and we are done. (cid:3) A standard application of this result is the construction of the skew-field of fractions of a freealgebra.
Example 6.14.
Let k h S i , the free algebra on a set S . Then there exists a unique homologicalepimorphism A → K ( S ) where K ( S ) is a skew-field. Indeed, A is a hereditary algebra with K ( A ) = Z , thus there exists a unique projective rank on A that uniquely extends to a rank ρ function on Perf ( A ). The corresponding derived localization L ρ ( A ) (which coincides with theCohn-Schofield localization since A is hereditary) is therefore a skew-field, commonly known asthe free field on S , [Coh95]. .2. Derived fields of fractions.
Given an ordinary ring A we will understand a classical field of fractions of A to be a (nonderived) localization A → K where K is a skew-field. Aderived field of fractions of A is a dg skew-field K together with a derived localization map A → K . Remark 6.15.
This definition of a classical field of fractions agrees with the standard notionthat ordinarily assumes that A is a commutative domain. In the noncommutative case thedefinition accepted e.g. in [Coh95] is different, in particular the map A → K is assumed to bean embedding. Our definition is one that extends most naturally to the dg context.Corollary 6.5 classifies derived fields of fractions of a ring A in terms of localizing rankfunctions on Perf ( A ). The following examples demonstrate the existence of such (genuinelyderived) fields of fractions and thus, localizing rank functions (that cannot be reduced to classicalSylvester rank functions) even for finite-dimensional algebras over fields. Example 6.16. (1) Let A = k [ M ] be the 5-dimensional algebra of Example 5.27. We saw that there is aderived localization map k [ M ] → k [ x ] where | x | = 1. Composing this with inverting x , we obtain a derived localization map k [ M ] → k [ x, x − ] where the target is a gradedskew-field of period 1, i.e. a derived fraction field of A . The ‘classical’ part of this map isthe augmentation map k [ M ] → k ; this is an underived localization and thus, a classicalfraction field in the sense understood above.(2) This example is essentially contained in [CL20, Example 6.13]. Let k be a field ofcharacteristic 3 and consider A := k [ S ], the group algebra of the symmetric group on3 symbols. Then A contains an idempotent e such that L e A is (quasi-isomorphic to)the graded algebra generated by two indeterminates y and z with | z | = 2 , | y | = 3 and z = y . The latter algebra has a fraction field k [ x, x − ] with x = z − y so | x | = 1 .Thus, k [ x, x − ] is a derived fraction skew-field for A . Again, the ‘classical’ part of thelatter derived fraction field is the augmentation map of the group ring k [ S ]. References [Bal05] Paul Balmer. The spectrum of prime ideals in tensor triangulated categories.
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Email address : [email protected]
University of Lancaster, Department of Mathematics and Statistics, Lancaster, LA1 4YF, UK.
Email address : [email protected]@lancaster.ac.uk