Derived, coderived, and contraderived categories of locally presentable abelian categories
aa r X i v : . [ m a t h . C T ] J a n DERIVED, CODERIVED, AND CONTRADERIVED CATEGORIESOF LOCALLY PRESENTABLE ABELIAN CATEGORIES
LEONID POSITSELSKI AND JAN ˇS ˇTOV´I ˇCEK
Abstract.
For a locally presentable abelian category B with a projective gener-ator, we construct the projective derived and contraderived model structures onthe category of complexes, proving in particular the existence of enough homotopyprojective complexes of projective objects. We also show that the derived category D ( B ) is generated, as a triangulated category with coproducts, by the projectivegenerator of B . For a Grothendieck abelian category A , we construct the injec-tive derived and coderived model structures on complexes. Assuming Vopenka’sprinciple, we prove that the derived category D ( A ) is generated, as a triangulatedcategory with products, by the injective cogenerator of A . More generally, we de-fine the notion of an exact category with an object size function and prove that thederived category of any such exact category with exact κ -directed colimits of chainsof admissible monomorphisms has Hom sets. In particular, the derived category ofany locally presentable abelian category has Hom sets. Contents
Introduction 1Acknowledgement 41. Cotorsion Pairs 42. Weak Factorization Systems 73. Small Object Argument 104. Abelian Model Structures 145. Categories of Complexes 176. Locally Presentable Abelian Categories with a Projective Generator 197. Contraderived Model Structure 248. Grothendieck Abelian Categories 299. Coderived Model Structure 3310. Exact Categories with an Object Size Function 36References 43
Introduction
The definition of the unbounded derived category of an abelian category goes backto the work of Grothendieck and Verdier in 1960’s [52, 53], but efficient techniques ofworking with such derived categories started to be developed only in the 1988 paper f Spaltenstein [48] (who attributes the idea to J. Bernstein). Subsequently they wereextended to the derived categories of DG-modules by Keller [24] (see also [7]).The problem is that the derived category D + ( A ) of bounded below complexes inan abelian category A with enough injective objects is equivalent to the homotopycategory of bounded below complexes of injective objects K + ( A inj ), and similarly, thederived category D − ( B ) of bounded above complexes in an abelian category B withenough projective objects is equivalent to the homotopy category of bounded abovecomplexes of projective objects K − ( B proj ). Such triangulated equivalences are used inthe constructions of derived functors acting between bounded above or below derivedcategories. But these equivalences fail for unbounded derived categories, generallyspeaking. So one needs to use what came to be known as “homotopy injective” or“homotopy projective” resolutions for unbounded complexes.It took another decade or two to realize the existence and importance of an al-ternative point of view on unbounded complexes, in which arbitrary complexes ofinjective or projective objects are used as resolutions. The equivalence relation onthe unbounded complexes then needs to be modified accordingly; so the conventionalquasi-isomorphism is replaced by a finer equivalence relation, making the resulting“exotic derived category” larger than the conventional one.The homotopy category of unbounded complexes of projective modules was firstconsidered by Jørgensen [23], and the homotopy category of unbounded complexesof injective objects in a locally Noetherian Grothendieck category was first studiedby Krause [26]. Constructions realizing such triangulated categories, similarly to theconventional derived category, as quotient categories of the homotopy category of theambient abelian category, were emphasized in the memoir [35], where the terminologyof the derived categories of the first and second kind was suggested. In fact, thereare several “derived categories of the second kind”: at the bare minimum, one has todistinguish between the coderived and the contraderived category.In subsequent publications, two approaches to derived categories of the secondkind emerged. On the one hand, there is an elementary construction of coderivedand contraderived (as well as the so-called absolute derived categories) as certainquotient categories of the homotopy categories. It goes back to the book [34] and thememoir [35], and was developed further in the papers [14, 43, 38]. This approach isapplicable to a wide class of abelian categories, as well as to Quillen exact categories,exact DG-categories (such as categories of curved DG-modules), etc.On the other hand, there is an approach based on the set-theoretic techniquesof contemporary model category theory (essentially, the small object argument). Itgoes back to the papers [23, 26], and was developed by Becker [4] in the context ofcurved DG-modules over curved DG-rings. Other relevant papers in this directioninclude [32, 33] and [51]. For categories suited for applicability of set-theoreticalmethods, such as the categories of modules over associative rings, this approachleads to stronger and more general results that the elementary one. It is still an openquestion whether the two approaches agree, e. g., for module categories. n this paper, we follow the approach of Becker, generalizing it from module cate-gories to some locally presentable abelian categories. In fact, both derived categories(or abelian model structures) of the first and the second kind were considered in thepaper [4], and we also work out both of these in this paper. On the other hand,we do not consider (curved or uncurved) DG-structures, restricting ourselves to thecategories of complexes in abelian categories.One of the aims of this paper is to emphasize the duality-analogy between twonatural classes of abelian categories. On the one hand, there are the Grothendieckabelian categories . Grothendieck [20] proved that they have enough injective objects.Now it is known that any Grothendieck abelian category has enough homotopy in-jective complexes [2], and even enough homotopy injective complexes of injectiveobjects [46, 18]. Following in the footsteps of Becker [4], we construct the injectivederived and the coderived abelian model structures on the category C ( A ) of com-plexes in A . We also show that, assuming Vopˇenka’s principle, the derived category D ( A ) of a Grothendieck abelian category A is generated, as a triangulated categorywith products, by any injective cogenerator of A .On the other hand, there is a much less familiar, but no less natural class of locally presentable abelian categories B with enough projective objects [41, 39, 42].Various contramodule categories [34, 37, 3, 40] are representatives of this class. Weconstruct the projective derived and the contraderived abelian model structures onthe category C ( B ) of complexes in B . In particular, it follows that there are enoughhomotopy projective complexes of projective objects in B . We also show that thederived category D ( B ) is generated, as a triangulated category with coproducts, byany projective generator of B .Furthermore, the contraderived category D ctr ( B ) (in the sense of Becker) is equiv-alent to the homotopy category K ( B proj ) of complexes of projective objects in B . Forany locally presentable additive category E and a fixed object M ∈ E , there exists a(unique) locally presentable abelian category B with enough projective objects suchthat the full subcategory B proj ⊂ B of projective objects in B is equivalent to thefull subcategory Add ( M ) ⊂ E of direct summands of coproducts of copies of M in E [42, 43, 39]. So the homotopy category K ( Add ( M )) of complexes in E with the termsin Add ( M ) can be interpreted as the contraderived category D ctr ( B ). It follows fromour results that the homotopy category K ( Add ( M )) is a well-generated triangulatedcategory in the sense of Neeman [31, 25].More generally, we consider the derived category D ( E ) of an arbitrary locally pre-sentable abelian category E . Such categories may have neither injective nor projectiveobjects, and neither infinite direct sum nor infinite product functors in E need to beexact. So one cannot speak of the coderived or contraderived categories of E , andmodel category methods in the spirit of [4] do not seem to be applicable. Still weprove something about the derived category D ( E ), namely, that it has Hom sets (inother words, the derived category “exists” in the same universe in which E is a cate-gory). In fact, our most general result in this direction is applicable to a certain classof exact categories E (in the sense of Quillen). n Sections 1–5 we review and collect the preparatory material on cotorsion pairs,weak factorization systems, and abelian model structures, following mostly the pa-pers [22], [4], and [41]. In Sections 6–7, we consider a locally presentable abeliancategory B with enough projective objects. In Section 6, we construct the projectivederived model category structure on the category of complexes C ( B ), and in Section 7we produce the contraderived model category structure on C ( B ). In Section 8–9, weconsider a Grothendieck abelian category A . In Section 8, we work out the injec-tive derived model category structure on C ( A ), and in Section 9, we construct thecoderived model category structure on C ( A ).In all the four cases, we obtain a hereditary abelian (hence stable) combinatorialmodel category. Consequently, it follows that the related derived, contraderived, andcoderived categories are well-generated triangulated categories [44, 11]. These resultsare known for Grothendieck abelian categories [27, 33, 28]. So we include these inour discussion mostly for the sake of completeness of the exposition, and in order todemonstrate a uniform approach, making the duality-analogy between the abeliancategories A and B visible.Finally, in the last Section 10 we show that the derived category D ( E ) “exists” (or ina different terminology, has Hom sets, rather than classes) for any locally presentableabelian category E . Our results in this direction also apply to Quillen exact categorieswith a cardinal-valued object size function and exact κ -directed colimits of chains ofadmissible monomorphisms for a large enough regular cardinal κ . Acknowledgement.
The first-named author is supported by the GA ˇCR project 20-13778S and research plan RVO: 67985840. The second-named author is supportedby the GA ˇCR project 20-13778S.1.
Cotorsion Pairs
Let E be an abelian category and L , R ⊂ E be two classes of objects (equivalently,full subcategories) in E . The class L is said to be generating if every object of E is aquotient object of an object from L . Dually, the class R is said to be cogenerating ifevery object of E is a subobject of an object from R .We denote by L ⊥ ⊂ E the class of all objects X ∈ E such that Ext E ( L, X ) = 0 forall L ∈ L . Dually, we let ⊥ R ⊂ E denote the class of all objects Y ∈ E such thatExt E ( Y, R ) = 0 for all R ∈ R . A pair of classes of objects ( L , R ) in E is said to bea cotorsion pair if R = L ⊥ and L = ⊥ R . The class L is then always closed underdirect summands and all coproducts which exist in E , and dually R is always closedunder direct summands and products (regardless of exactness properties of productsand coproducts; see [12, Corollary 8.3] or [13, Corollary A.2]).For any class of objects S ⊂ E , the pair of classes R = S ⊥ and L = ⊥ R ⊂ E is acotorsion pair. The cotorsion pair ( L , R ) is said to be generated by the class S .Dually, for any class of objects T ⊂ E , the pair of classes L = ⊥ T and R = L ⊥ isa cotorsion pair. The cotorsion pair ( L , R ) is said to be cogenerated by the class T . cotorsion pair ( L , R ) in E is said to admit special precover sequences if for everyobject E ∈ E there exists a short exact sequence(1) 0 −−→ R ′ −−→ L −−→ E −−→ E with R ′ ∈ R and L ∈ L . The cotorsion pair ( L , R ) is said to admit specialpreenvelope sequences if for every object E ∈ E there exists a short exact sequence(2) 0 −−→ E −−→ R −−→ L ′ −−→ E with R ∈ R and L ′ ∈ L . The approximation sequences is a generic name for thespecial precover and special preenvelope sequences. Lemma 1.1.
Let ( L , R ) be a cotorsion pair in E such that the class L is generatingand the class R is cogenerating in E . Then the pair of classes ( L , R ) admits specialprecover sequences if and only if it admits special preenvelope sequences.Proof. This is a categorical generalization of the Salce lemmas [45]. The key obser-vation is that, in any cotorsion pair ( L , R ), both the classes L and R are closed underextensions in E . Suppose, e. g., that ( L , R ) admits special preenvelope sequences,and let E ∈ E be an object. By assumption, E is a quotient object of an object M ∈ L ; so there is a short exact sequence 0 −→ F −→ M −→ E −→ E . Let0 −→ F −→ R −→ L ′ −→ F ∈ E ;so R ∈ R and L ′ ∈ L . Taking the pushout of the pair of monomorphisms F −→ M and F −→ R produces a special precover sequence 0 −→ R −→ L −→ E −→ L ∈ L is an extension of the objects M and L ′ . (cid:3) A cotorsion pair ( L , R ) is E is said to be complete if it admits both special precoverand special preenvelope sequences.Whenever there are enough projective objects in E , the class L in any cotorsionpair ( L , R ) is generating, because it contains all the projective objects. It followsthat any cotorsion pair ( L , R ) admitting special preenvelope sequences is complete.Dually, whenever there are enough injective objects in E , the class R in any cotorsionpair ( L , R ) is cogenerating, because it contains all the injective objects. It followsthat any cotorsion pair ( L , R ) admitting special precover sequences is complete.When there are both enough projectives and injectives in E , the assumption ofLemma 1.1 holds automatically, so a cotorsion pair admits special precover sequencesif and only if it admits special preenvelope sequences. Such is the situation in theabelian categories of modules over associative rings.For any class of objects F ⊂ E , we denote by F ⊕ the class of all direct summandsof objects from F . Lemma 1.2.
Let ( L , R ) be a pair of classes of objects in E such that Ext E ( L, R ) = 0 for all L ∈ L and R ∈ R . Assume that the approximation sequences (1–2) with R , R ′ ∈ R and L , L ′ ∈ L exist for all objects E ∈ E . Then ( L ⊕ , R ⊕ ) is a completecotorsion pair in E . (cid:3) Proof.
This is an analogue of Lemma 2.1 below. It suffices to show that L ⊥ ⊂ R ⊕ and ⊥ R ⊂ L ⊕ . Indeed, let E ∈ E be an object belonging to ⊥ R . By assumption, here exists a short exact sequence 0 −→ R ′ −→ L −→ E −→ E with R ′ ∈ R and L ∈ L . Now Ext E ( E, R ′ ) = 0, hence E is a direct summand of L . (cid:3) Let ( f ij : F i → F j ) ≤ i Proposition 1.3. For any class of objects R ⊂ E , the class of objects ⊥ R ⊂ E isclosed under transfinitely iterated extensions (in the sense of the directed colimit). Inother words, we have ⊥ R = Fil ( ⊥ R ) . (cid:3) Proof. This is [41, Lemma 4.5]. Alternatively, the assertion can be deduced fromLemma 2.2, Proposition 2.3(a), and Proposition 2.5(a) below. (cid:3) Lemma 1.4. Let ( L , R ) be a cotorsion pair in E such that the class L is generatingand the class R is cogenerating in E . Then the following conditions are equivalent: (1) the class L contains the kernels of epimorphisms between its objects; (2) the class R contains the cokernels of monomorphisms between its objects; (3) Ext E ( L, R ) = 0 for all L ∈ L and R ∈ R ; (4) Ext n E ( L, R ) = 0 for all L ∈ L , R ∈ R , and n ≥ .Proof. Use the long exact sequences of Ext groups associated with short exact se-quences of objects in E in order to prove the equivalences (1) ⇐⇒ (3) ⇐⇒ (2). Thendeduce (4) from either (1) or (2). Details can be found in [50, Lemma 6.17] or [47,Lemma 4.25]. (cid:3) otice that the assumption of Lemma 1.4 holds, in particular, for any complete co-torsion pair ( L , R ). A cotorsion pair satisfying the equivalent conditions of Lemma 1.4is said to be hereditary .2. Weak Factorization Systems Let E be a category. One says that an object A ∈ E is a retract of an object B ∈ E if there exist morphisms i : A −→ B and p : B −→ A such that pi = id A isthe identity endomorphism. A morphism f : A −→ B in E is said to be a retract ofa morphism g : C −→ D in E if f is a retract of g as objects of the category E → ofmorphisms in E (with the commutative squares in E being the morphisms in E → ).Given a class of morphisms F in E , we denote by F the class of all the retracts ofmorphisms from F in E .Let l : A −→ B and r : C −→ D be two morphisms in E . One says that r hasthe right lifting property with respect to l or, which is the same, l has the left liftingproperty with respect to r , if for every pair of morphisms f : A −→ C and g : B −→ D such that rf = gl there exists a morphism t : B −→ C such that f = tl and g = rt . Inother words, any commutative square as in the diagram can be filled with a diagonalarrow making both the triangles commutative: A / / l (cid:15) (cid:15) C r (cid:15) (cid:15) B / / ? ? ⑦⑦⑦⑦⑦⑦⑦ D One can easily check that if a morphism r has the right lifting property with respectto a morphism l , then any retract of r has the right lifting property with respect toany retract of l .Let L and R be two classes of morphisms in E . We will denote by L (cid:3) the class ofall morphisms x in E having the right lifting property with respect to all morphisms l ∈ L . Similarly, we let (cid:3) R denote the class of all morphisms y in E having the leftlifting property with respect to all morphisms r ∈ R .Let ( L , R ) be a pair of classes of morphisms in E . We will say that the pair ( L , R ) has the lifting property if R ⊂ L (cid:3) , or equivalently, L ⊂ (cid:3) R . Furthermore, we will saythat the pair ( L , R ) has the factorization property if every morphism f in E can bedecomposed as f = rl with l ∈ L and r ∈ R . A pair of classes of morphisms ( L , R )in E is called a weak factorization system if R = L (cid:3) , L = (cid:3) R , and the pair ( L , R )has the factorization property. Lemma 2.1. Let ( L , R ) be a pair of classes of morphisms in E having the liftingand factorization properties. Then the pair of classes of morphisms ( L , R ) is a weakfactorization system in E . roof. In view of the above discussion of the lifting properties of retracts, we onlyneed to check the inclusions L (cid:3) ⊂ R and (cid:3) R ⊂ L . Indeed, let f = rl be a compositionof two morphisms in E with l ∈ L and r ∈ R . One observes that if f has the rightlifting property with respect to l , then f is a retract of r ; and similarly, if f has the leftlifting property with respect to r , then f is a retract of l ; see [21, Lemma 1.1.9]. (cid:3) Let ( f ij : E i → E j ) ≤ i For any class of morphisms R in E , the class of morphisms (cid:3) R isclosed under pushouts, transfinite compositions (in the sense of the directed colimit),coproducts and retracts. In other words, we have (cid:3) R = C of ( (cid:3) R ) . (cid:3) Now let E be an abelian category. Let L and R ⊂ E be two classes of objects. Amorphism in E is said to be an L -monomorphism if it is a monomorphism whose coker-nel belongs to L . Similarly, a morphism in E is said to be an R -epimorphism if it is anepimorphism whose kernel belongs to R . We denote the class of all L -monomorphismsby L - M ono and the class of all R -epimorphisms by R - E pi . Proposition 2.3. (a) The inclusion R - E pi ⊂ L - M ono (cid:3) holds if and only if R ⊂ L ⊥ . (b) Moreover, for any class of objects L ⊂ E , the kernel of any morphism from L - M ono (cid:3) belongs to L ⊥ . (c) If the class L ⊂ E is generating, then the class L - M ono (cid:3) consists of epimor-phisms. Consequently, L - M ono (cid:3) = L ⊥ - E pi in this case.Proof. This is [41, Lemmas 3.1 and 4.2]. For later use, we note that in order to provein part (c) that any r ∈ L - M ono (cid:3) is an epimorphism, it suffices to use that r has theright lifting property with respect to all 0 −→ L , L ∈ L . (cid:3) otice that the class L - M ono (cid:3) does not always consist of epimorphisms. To givea trivial example, if L = ∅ or L = { } , then all the L -monomorphisms are isomor-phisms, hence all the morphisms in E belong to L - M ono (cid:3) (cf. Examples 3.6 below).However, if the class L - M ono (cid:3) consists of epimorphisms, then it is clear from Propo-sition 2.3(a–b) that L - M ono (cid:3) = L ⊥ - E pi .A weak factorization system ( L , R ) in an abelian category E is said to be abelian if the class L consists of monomorphisms, the class R consists of epimorphisms, amonomorphism l : A −→ B belongs to L if and only if the morphism 0 −→ coker( l )belongs to L , and an epimorphism r : C −→ D belongs to R if and only if themorphism ker( r ) −→ R . In other words, a weak factorization system( L , R ) is abelian if and only if there exists a pair of classes of objects ( L , R ) in E suchthat L = L - M ono and R = R - E pi . Theorem 2.4. Let ( L , R ) be a pair of classes of objects in an abelian category E .Then the pair of classes of morphisms L = L - M ono and R = R - E pi forms a weakfactorization system in E if and only if ( L , R ) is a complete cotorsion pair. So abelianweak factorization systems correspond bijectively to complete cotorsion pairs in E .Proof. This result is essentially due to Hovey [22]. “Only if”: it is clear from Propo-sition 2.3(a) that the equation R - E pi = L - M ono (cid:3) implies R = L ⊥ ; similarly, theequation L - M ono = (cid:3) ( R - E pi ) implies L = ⊥ R .To prove existence of special precover sequences for ( L , R ), consider an object E ∈ E . Since the pair of classes ( L , R ) has the factorization property by assumption,the morphism 0 −→ E can be factorized as 0 −→ M −→ E , where the morphism l : 0 −→ M belongs to L and the morphism r : M −→ E belongs to R . Now we have M ∈ L and the morphism r : M −→ E is an epimorphism with the kernel belongingto R . The dual argument proves existence of special preenvelope sequences.“If”: If ( L , R ) is a complete cotorsion pair, then the class L is generating and theclass R is cogenerating. By Proposition 2.3(c) and its dual version, it follows that L - M ono (cid:3) = R - E pi and (cid:3) ( R - E pi ) = L - M ono .Now let f : A −→ B be a morphism in E . We want to decompose f into an L -monomorphism followed by R -epimorphism. We follow an argument from [34, Sec-tion 9.1] (for the classical approach, see [22, Proposition 5.4]). Choose an object P ∈ L together with a morphism p : P −→ B such that the morphism ( f, p ) : A ⊕ P −→ B is an epimorphism. (E. g., one can choose p to be an epimorphism; alternatively,when f is an epimorphism, one can take P = 0.) Denote by K the kernel of the mor-phism ( f, p ), and choose a special preenvelope sequence 0 −→ K −→ R −→ L ′ −→ K ∈ E (so R ∈ R and L ′ ∈ L ).Denote the monomorphism K −→ A ⊕ P by ( a, q ) and the monomorphism K −→ R by k . Let C be the cokernel of the monomorphism ( a, q, k ) : K −→ A ⊕ P ⊕ R .Consider the morphism ( f, p, 0) : A ⊕ P ⊕ R −→ B . The composition ( f, p, ◦ ( a, q, k ) = ( f, p ) ◦ ( a, q ) vanishes; so the morphism ( f, p, 0) factorizes through theepimorphism A ⊕ P ⊕ R −→ C , providing a morphism r : C −→ B . Denote by l : A −→ C the composition of the coproduct inclusion (id A , , 0) : A −→ A ⊕ P ⊕ R with the epimorphism A ⊕ P ⊕ R −→ C . By construction, we have rl = f . inally, the morphism r is an epimorphism with the kernel ker( r ) = R , hence r ∈ R - E pi . The morphism l is an a monomorphism whose cokernel is the middleterm of a short exact sequence 0 −→ P −→ coker( l ) −→ L ′ −→ 0. As both theobjects P and L ′ belong to L and the class L is closed under extensions, we canconcude that l ∈ L - M ono .Notice that the above construction is not self-dual; one could also proceed in thedual way, choosing an object J ∈ R together with a morphism j : A −→ J suchthat ( f, j ) : A −→ B ⊕ J is a monomorphism, considering the cokernel of ( f, j ), etc.(see [34, Section 9.2]). (cid:3) Proposition 2.5. Let S be a class of objects in an abelian category E . Then (a) any Fil ( S ) -monomorphism in E belongs to C ell ( S - M ono ) ; (b) the cokernel of any morphism from C ell ( S - M ono ) belongs to Fil ( S ) .Proof. This is [41, Lemma 4.6(a–b) and Remark 4.7]. Part (b) is straightforward;part (a) is more involved. (cid:3) Small Object Argument Let λ be an infinite cardinal. A poset I is said to be λ -directed if any subset J ⊂ I of the cardinality less than λ has an upper bound in I , i. e., an element i ∈ I suchthat j ≤ i for all j ∈ J . In particular, I is ω -directed (where ω denotes the cardinalof nonnegative integers) if and only if I is directed in the usual sense.A λ -directed colimit in a category E is the colimit of a diagram indexed by a λ -directed poset. Assuming that all the λ -directed colimits exist in E , an object E ∈ E is said to be λ -presentable if the functor Hom E ( E, − ) : E −→ Sets preserves λ -directed colimits.Let λ be a regular infinite cardinal. A category E is said to be λ -accessible if allthe λ -directed colimits exist in E and there is a set of λ -presentable objects G ⊂ E such that every object in E is a λ -directed colimit of objects from G . A cocomplete λ -accessible category is said to be locally λ -presentable .Equivalently, a cocomplete category E is locally λ -presentable if and only if it has astrongly generating set of λ -presentable objects [1, Theorem 1.20]. We do not definewhat it means for a set of generators in a category to be a set of strong generators(see [1, Section 0.6] for the discussion), as we are only interested in locally presentable abelian categories in this paper. In an abelian category, any set of generators is a setof strong generators.A category is said to be accessible if it is λ -accessible for some regular cardinal λ .Similarly, a category is said to be locally presentable if it is locally λ -presentable forsome regular cardinal λ .The following theorem summarizes Quillen’s classical “small object argument”.For a more general formulation, see [21, Theorem 2.1.14] or [47, Proposition 2.1]. heorem 3.1. Let E be a locally presentable category and S be a set of morphismsin E . Then the pair of classes of morphisms C ell ( S ) and S (cid:3) has the factorizationproperty. Consequently, the pair of classes of morphisms C of ( S ) and S (cid:3) is a weakfactorization system. Moreover, the factorization f = rl of an arbitrary morphism f in E into the composition of a morphism r ∈ S (cid:3) and l ∈ C ell ( S ) can be chosen sothat it depends functorially on f .Proof. See, e. g., [5, Proposition 1.3]. The assertion of the theorem can be strength-ened, in particular, by defining the class C of ( S ) in a more restrictive way (this in-volves formulating a correspondingly adjusted version of Lemma 2.1); see [5, Defini-tion 1.1(ii)]. (cid:3) Proposition 3.2. Let λ be a regular cardinal, E be a locally λ -presentable abeliancategory, and S be a set of λ -presentable objects in E . Then any S -monomorphismin E is a pushout of an S -monomorphism with a λ -presentable codomain.Proof. This is [41, Lemma 3.4]. (cid:3) The next two theorems extend the classical Eklof–Trlifaj theorem about the co-torsion pair generated by a set of modules [15, Theorem 10] to the realm of locallypresentable abelian categories. Given an additive category E and a class of objects S ⊂ E , we denote by Add ( S ) = Add E ( S ) ⊂ E the class of all direct summands ofcoproducts of copies of objects from S in E . Theorem 3.3. Let E be a locally presentable abelian category and ( L , R ) be a cotor-sion pair generated by a set of objects in E . Assume that the class L is generatingand the class R is cogenerating in E . Then the cotorsion pair ( L , R ) is complete.Proof. This is [41, Theorem 3.5]. Let S be a set of objects in E such that R = S ⊥ .Using the assumption that the class L is generating, one can construct a possiblylarger set of objects S ⊂ E , S ⊂ S , such that R = S ⊥ and every object of E is aquotient object of an object from Add ( S ).Let λ be a regular cardinal such that the category E is locally λ -presentable andall the objects in S are λ -presentable (this is always possible, see [1, Remark belowTheorem 1.20]). Applying Theorem 3.1 to the set S of (representatives of isomorphismclasses of) all S -monomorphisms with λ -presentable codomains in E , we conclude thepair of classes C of ( S ) and S (cid:3) forms a weak factorization system in E .By Proposition 3.2 and Lemma 2.2, we have S (cid:3) = S - M ono (cid:3) and we also deducethat each r ∈ S - M ono (cid:3) has the right lifting property with respect to all morphismsof the form 0 −→ L , L ∈ Add ( S ). Using the fact that the class Add ( S ) is generatingin E and following the proof of Proposition 2.3(c), one can see that S - M ono (cid:3) = R - E pi . The dual assertion to Proposition 2.3(c) tells that (cid:3) ( R - E pi ) = L - M ono , dueto the assumption that the class R is cogenerating. Thus we have S (cid:3) = R - E pi and C of ( S ) = L - M ono , so the pair of classes L - M ono and R - E pi is a weak factorizationsystem, and the desired assertion follows from Theorem 2.4. (cid:3) heorem 3.4. Let E be a locally presentable abelian category and ( L , R ) be the co-torsion pair generated by a set of objects S in E . Assume that the class of objects Fil ( S ) is generating in E . Then one has L = Fil ( S ) ⊕ .Proof. This is a particular case of [41, Theorem 4.8(d)]. By Proposition 2.5(a),we have Fil ( S )- M ono ⊂ C ell ( S - M ono ). In view of Lemma 2.2, it follows that Fil ( S )- M ono (cid:3) = S - M ono (cid:3) . Applying Proposition 2.3(c), we conclude that theclass S - M ono (cid:3) consists of epimorphisms. By Proposition 2.3(a–b), it follows that S - M ono (cid:3) = S ⊥ - E pi = R - E pi .Let λ be a regular cardinal such that the category E is locally λ -presentable and allthe objects in S are λ -presentable. Applying Theorem 3.1 to the S of (representativesof isomorphism classes of) all S -monomorphisms with λ -presentable codomains in E ,we conclude that every morphism in E factorizes as the composition of a morphismfrom C ell ( S ) followed by a morphism from S (cid:3) . By Proposition 3.2 and Lemma 2.2,we have S (cid:3) = S - M ono (cid:3) ; thus S (cid:3) = R - E pi .Given an object M ∈ E , consider the morphism 0 −→ M and decompose itas 0 −→ F −→ M so that the morphism 0 −→ F belongs to C ell ( S ) and themorphism F −→ M belongs to S (cid:3) . By Proposition 2.5(b), we have F ∈ Fil ( S ). Theargument above tells that F −→ M is an epimorphism with a kernel R ∈ R . We haveconstructed a special precover (short exact) sequence 0 −→ R −→ F −→ M −→ M with F ∈ Fil ( S ) and R ∈ R . Now if M ∈ L , then Ext E ( M, R ) = 0, hence M isa direct summand of F . (cid:3) Remark 3.5. For any cotorsion pair ( L , R ) in an abelian category E , the class L contains all the projective objects in E and the class R contains all the injective objectsin E . Hence the assumption that the class L is generating holds automatically if E hasenough projectives, and the assumption that R is cogenerating holds automatically if E has enough injectives. So Theorem 3.3 is usually stated without these assumptionsfor module categories E , or more generally, for Grothendieck abelian categories E with enough projective objects. For categories without enough projective/injectiveobjects, however, Theorem 3.3 does not hold without such assumptions about theclasses L and R , as the following counterexamples demonstrate. Examples 3.6. (1) Let p be a prime number and A be the category of p -primarytorsion abelian groups, that is abelian groups A such that for every a ∈ A there exists n ≥ p n a = 0 in A . Then A is a locally finitely presentable Grothendieckabelian category. Let S be the empty set of objects in A , or alternatively, if onewishes, let S be the set consisting of the zero object only, S = { } ⊂ A .Then the class R = S ⊥ ⊂ A coincides with the whole category A , and the class L = ⊥ R consists of all the projective objects in A , which means the zero object only.Indeed, for any abelian group A ∈ A , one has Ext A ( A, Z /p Z ) ≃ Hom A ( p A, Z /p Z ) = 0if A = 0, where p A ⊂ A denotes the subgroup of all elements annihilated by p in A .Thus we have L = { } and R = A , which is clearly not a complete cotorsion pair in A (the special precover sequences do not exist). t is instructive to consider the classes L = C of ( S ) and R = S (cid:3) from the proofof Theorem 3.4 in this case. The set S of representatives of isomorphism classes of S -monomorphisms with finitely presentable (or λ -presentable) codomains consists ofisomorphisms; so R is the class of all morphisms in A . It follows that L is the classof all isomorphisms in A .(2) Let p be a prime number and B be the category of p -contramodule abeliangroups, that is abelian groups B such that Hom Z ( Z [ p − ] , B ) = 0 = Ext Z ( Z [ p − ] , B ).Then B is a locally ℵ -presentable abelian category with enough projective objects [41,Example 4.1(3)]. In fact, the group of p -adic integers P = Z p is a projective generatorof B . Let S = { P, S } be the set consisting of the projective generator P and thesimple abelian group S = Z /p Z , or alternatively, if one wishes, let S = { S } be theset consisting of the group S only.Then the class R = S ⊥ ⊂ B consists of the zero object only. Indeed, for any abeliangroup B ∈ B , one has Ext B ( Z /p Z , B ) ≃ B/pB = 0 if B = 0 (as B = pB would implysurjectivity of the natural map Hom Z ( Z [ p − ] , B ) −→ B ) [41, Example 4.1(4)]. Hencethe class L = ⊥ R coincides with the whole category B . Thus we have L = B and R = { } , which is clearly not a complete cotorsion pair in B (the special preenvelopesequences do not exist).Notice that Theorem 3.4 is not applicable in this example with S = { S } , but it be-comes applicable if one takes S = { P, S } . So one can conclude that B = Fil ( { P, S } ) ⊕ .It is instructive to consider the classes L = C of ( S ) and R = S (cid:3) from the proofsof Theorems 3.3 and 3.4 in this case. Take S = { P, S } ; then, following the proof ofTheorem 3.4, R = R - E pi is the class of all epimorphisms in B with the kernels in R = { } , which means that R is the class of all isomorphisms in B . Hence L is theclass of all morphisms in B .A weak factorization system ( L , R ) in E is said to be cofibrantly generated if thereexists a set of morphisms S in E such that R = S (cid:3) . Lemma 3.7. Let E be a locally presentable abelian category and ( L , R ) be an abelianweak factorization system in E . Let ( L , R ) be the corresponding complete cotorsionpair in E , as in Theorem 2.4; so L = L - M ono and R = R - E pi . Then the weakfactorization system ( L , R ) is cofibrantly generated if and only if the cotorsion pair ( L , R ) is generated by a set of objects.Proof. This is essentially a generalization of [4, Proposition 1.2.7]. The “if” impli-cation is clear from the proof of Theorem 3.3. To prove the “only if”, let S be aset of morphisms in E such that R = S (cid:3) . Then S ⊂ L - M ono . Denote by S the set(of representatives of isomorphism classes) of cokernels of all the morphisms from S .Then S ⊂ S - M ono ⊂ L - M ono , hence R = S - M ono (cid:3) . As the class R consists ofepimorphisms, one can see from Lemma 2.3(a–b) that R = S ⊥ . (cid:3) . Abelian Model Structures Abstracting from the definition of derived categories, if one is given a category E and a class of morphisms W in E , one often wishes to understand the category E [ W − ],where one freely adds inverses to all morphisms in W . This is analogous to localizationin commutative algebra and can be achieved by a similar construction [16, § I.1], butunlike in commutative algebra it is in general extremely difficult to understand what E [ W − ] looks like. The topologically motivated notion of model category solves thisproblem (see [21, Theorem 1.2.10]) at the cost of requiring more structure than justthe pair ( E , W ).A model structure on a category E is a triple of classes of morphisms L , R , and W satisfying the following conditions:(i) the pair of classes L and R ∩ W is a weak factorization system;(ii) the pair of classes L ∩ W and R is a weak factorization system;(iii) the class W is closed under retracts and satisfies the two-out-of-three property:for any composable pair of morphisms f and g in E , if two of the threemorphisms f , g , and gf belong to W , then the third one also does.Morphisms in the classes L , R , and W are called cofibrations , fibrations , and weakequivalences , respectively. Morphisms in the class L ∩ W are called trivial cofibrations ,and morphisms in the class R ∩ W are called trivial fibrations .A model category is a complete, cocomplete category with a model structure( L , W , R ), and E [ W − ] is called the homotopy category of the model category inthis context. We will be interested only in the so-called stable model categories.This condition implies that the homotopy category carries a natural triangulatedstructure; see [21, Chapter 7]. We will not use this condition directly, however, butonly through [4, Corollary 1.1.15 and the preceding discussion], which says that anyhereditary abelian model category (to be defined later in this section) is stable.In order to use the advantage of set-theoretic methods, one usually considers thefollowing technical, but widely satisfied conditions. A model structure ( L , W , R ) ona category E is said to be cofibrantly generated if both the weak factorization systems( L , R ∩ W ) and ( L ∩ W , R ) are cofibrantly generated. Cofibrantly generated modelcategories whose underlying category is locally presentable are called combinatorial (all locally presentable categories are complete and cocomplete).An important point is that the homotopy category of any stable combinatorialmodel category is well-generated triangulated in the sense of Neeman; see [44, Propo-sition 6.10] or [11, Theorems 3.1 and 3.9]. The theory of well-generated triangulatedcategories [31, 25] gained popularity because it is applicable to a wide range of natu-rally occurring triangulated categories and allows (to a somewhat limited extent) toobtain results analogous to consequences of the small object argument purely in thelanguage of triangulated categories, without a reference to any enhancement.In Sections 6–9, our aim will be to construct certain concrete stable combina-torial model structures on the categories of (unbounded) complexes in some lo-cally presentable abelian categories. It will follow that the corresponding homotopy ategories—in our case derived, coderived and contraderived categories of the abeliancategories—are well-generated triangulated.Let E be a category with a model structure. An object L ∈ E is said to be cofibrant if the morphism ∅ −→ L is a cofibration. An object R ∈ E is said to be fibrant if themorphism R −→ ∗ is a fibration. Here ∅ and ∗ denote the initial and the terminalobject in the category E , respectively (which we presume to exist). For any additivecategory E , they are the same: ∅ = 0 = ∗ .More generally, a category E is called pointed if it has an initial and a terminalobject, and they coincide. The initial-terminal object of a pointed category is calledthe zero object and denoted by 0. Given a pointed category E with a model structure,an object W ∈ E is said to be weakly trivial if the morphism 0 −→ W is a weakequivalence, or equivalently, the morphism W −→ trivially cofibrant , and weakly trivial fibrantobjects are said to be trivially fibrant .The next definition of an abelian model structure is due to Hovey [22] (see [19]for a generalization to exact categories the sense of Quillen). Let E be an abeliancategory. A model structure ( L , W , R ) on E is said to be abelian if L is the class ofall monomorphisms with cofibrant cokernels and R is the class of all epimorphismswith fibrant kernels. A model category is said to be abelian if its underlying categoryis abelian and the model structure is abelian. Lemma 4.1. In an abelian model structure, L ∩ W is the class of all monomorphismswith trivially cofibrant cokernels and R ∩ W is the class of all epimorphisms withtrivially fibrant kernels.Proof. This is a part of [22, Proposition 4.2]. Let L , R , and W denote the classesof cofibrant, fibrant, and weakly trivial objects in E , respectively. Then we have L = L - M ono and R = R - E pi . The class L - M ono (cid:3) = R ∩ W ⊂ R consists of epi-morphisms. By Proposition 2.3(a–b), it follows that L - M ono (cid:3) = L ⊥ - E pi . Similarly,the class (cid:3) ( R - E pi ) = L ∩ W ⊂ L consists of monomorphisms. The dual assertionsto Proposition 2.3(a–b) tell that (cid:3) ( R - E pi ) = ( ⊥ R )- M ono . Now it is clear that L ⊥ = R ∩ W is the class of all trivially fibrant objects and ⊥ R = L ∩ W is the classof all trivially cofibrant objects. (cid:3) A class of objects W ⊂ E is said to be thick if it is closed under direct summandsand, for any short exact sequence 0 −→ A −→ B −→ C −→ E , if two of thethree objects A , B , C belong to W then the third one also does. Theorem 4.2. Abelian model structures on an abelian category E correspond bijec-tively to triples of classes of objects ( L , W , R ) such that (1) the pair of classes L and R ∩ W is a complete cotorsion pair; (2) the pair of classes L ∩ W and R is a complete cotorsion pair; (3) W is a thick class.The correspondence assigns to a triple of classes of morphisms ( L , W , R ) the tripleof classes of objects ( L , W , R ) , where L is the class of all cofibrant objects, R is the lass of all fibrant objects, and W is the class of all weakly trivial objects in themodel structure ( L , W , R ) . Conversely, to a triple of classes of objects ( L , W , R ) thetriple of classes of morphisms ( L , W , R ) is assigned, where L = L - M ono , R = R - E pi , L ∩ W = ( L ∩ W )- M ono , R ∩ W = ( R ∩ W )- E pi , and W is the class of all morphisms w decomposable as w = rl , where l ∈ ( L ∩ W )- M ono and r ∈ ( R ∩ W )- E pi .Proof. This is a part of [22, Theorem 2.2] (for a further generalization, see [19, The-orem 3.3]). Given a model structure ( L , W , R ) on E such that L = L - M ono and R = R - E pi , we have L ∩ W = ( L ∩ W )- M ono and R ∩ W = ( R ∩ W )- E pi by Lemma 4.1.Hence the pairs of classes of objects ( L , R ∩ W ) and ( L ∩ W , R ) are complete cotorsionpairs by Theorem 2.4. The class of objects W ⊂ E is thick by [22, Lemma 4.3].Conversely, given a triple of classes of objects ( L , W , R ) in E satisfying (1–3), definethe triple of classes of morphisms ( L , W , R ) as stated in the theorem. Then the pairsof classes ( L , ( R ∩ W )- E pi ) and (( L ∩ W - E pi ) , R ) are weak factorization systems byTheorem 2.4. The equations L ∩ W = ( L ∩ W )- M ono and R ∩ W = ( R ∩ W )- E pi hold by [22, Lemma 5.8]. The class of morphisms W satisfies the two-out-of-threeproperty by [22, Proposition 5.12]. (cid:3) In the sequel, we will identify abelian model structures with the triples of classesof objects ( L , W , R ) using Theorem 4.2, and write simply “an abelian model structure( L , W , R ) on an abelian category E ”. Corollary 4.3. An abelian model structure ( L , W , R ) on a locally presentable abeliancategory E is cofibrantly generated if and only if both the cotorsion pairs ( L , R ∩ W ) and ( L ∩ W , R ) are generated by some sets of objects.Proof. Follows immediately from Lemma 3.7. (cid:3) Lemma 4.4. Let ( L , W , R ) be an abelian model structure on an abelian category E .Then the following conditions are equivalent: (1) the class L is closed under the kernels of epimorphisms in E ; (2) the class L ∩ W is closed under the kernels of epimorphisms in E ; (3) the class R is closed under the cokernels of monomorphisms in E ; (4) the class R ∩ W is closed under the cokernels of monomorphisms in E .Proof. The equivalences (1) ⇐⇒ (4) and (2) ⇐⇒ (3) hold by Lemma 1.4. The impli-cations (1) = ⇒ (2) and (3) = ⇒ (4) hold because the class of weakly trivial objects W is thick (so it is closed under both the kernels of epis and the cokernels of monos). (cid:3) An abelian model structure satisfying the equivalent conditions of Lemma 4.4 issaid to be hereditary .A model structure ( L , W , R ) on a category B is called projective if all the objectsof B are fibrant. For an abelian model structure, this means that R = B , or equiv-alently, L ∩ W = B proj is the class of all projective objects in B . In other words, anabelian model structure is projective if and only if the trivial cofibrations are themonomorphisms with projective cokernel. ually, a model structure ( L , W , R ) on a category A is called injective if all theobjects of A are cofibrant. For an abelian model structure, this means that L = A ,or equivalently, R ∩ W = A inj is the class of all injective objects in A . In other words,an abelian model structure is injective if and only if the trivial fibrations are theepimorphisms with injective kernel.It is clear from the definitions that all projective abelian model structures and allinjective abelian model structures are hereditary [4, Corollary 1.1.12]. (Indeed, anyprojective abelian model structure obviously satisfies the condition (3) of Lemma 4.4,while any injective abelian model structure obviously satisfies the condition (1).) Lemma 4.5 ([4, Corollary 1.1.9]) . Let A and B be abelian categories. Then (a) a pair of classes of objects ( W , R ) in A defines an injective abelian model struc-ture ( A , W , R ) on A if and only if A has enough injective objects, ( W , R ) is a completecotorsion pair in A with R ∩ W = A inj , and the class W is thick. (b) a pair of classes of objects ( L , W ) in B defines a projective abelian model struc-ture ( L , W , B ) on B if and only if B has enough projective objects, ( L , W ) is a completecotorsion pair in B with L ∩ W = B proj , and the class W is thick.Proof. The assertions follow from Theorem 4.2. One only needs to observe [4] that( A , R ∩ W ) is a complete cotorsion pair in A if and only if A has enough injectivesand R ∩ W = A inj . Similarly, ( L ∩ W , B ) is a complete cotorsion pair in B if and onlyif B has enough projectives and L ∩ W = B proj . (cid:3) The idea of the following lemma can be traced back at least to [6, Theorem VI.2.1]. Lemma 4.6 ([4, Lemma 1.1.10]) . (a) Let ( W , R ) be a hereditary complete cotorsionpair in an abelian category A such that R ∩ W = A inj . Then the class of objects W isthick in A . (b) Let ( L , W ) be a hereditary complete cotorsion pair in an abelian category B suchthat L ∩ W = B proj . Then the class of objects W is thick in B .Proof. Part (a): in any hereditary cotorsion pair ( W , R ), the class of objects W isclosed under direct summands, extensions, and the kernels of epimorphisms. It re-mains to show that W is closed under the cokernels of monomorphisms in our as-sumptions. We follow the argument from [4]. Let 0 −→ W −→ V −→ U −→ A with W , V ∈ W . Then Ext A ( U, R ) = 0for all R ∈ R , since Ext A ( W, R ) = 0 = Ext A ( V, R ). Let us show that this impliesExt A ( U, R ) = 0. Choose a special precover sequence 0 −→ R ′ −→ J −→ R −→ R ∈ A with R ′ ∈ R and J ∈ W . Then J ∈ R ∩ W = A inj , since R ′ , R ∈ R .Hence Ext A ( U, R ) ≃ Ext A ( U, R ′ ) = 0. Part (b) is dual. (cid:3) Categories of Complexes Let E be an additive category. We denote by C ( E ) the category of (unbounded)complexes in E . The category C ( E ) is abelian whenever the category E is. We denoteby C • C • [1] the usual shift functor, where C • [1] i = C i +1 and d iC • [1] = − d i +1 C • . urthermore, let K ( E ) denote the homotopy category of complexes in E , that is,the additive quotient category of C ( E ) by the ideal of morphisms cochain homotopicto zero. Then K ( E ) is a triangulated category [52, 53].Let E be an abelian category. A short exact sequence of complexes 0 −→ A • −→ C • −→ B • −→ E is said to be termwise split if the short exact sequence0 −→ A i −→ C i −→ B i −→ E for every i ∈ Z .The next lemma is well-known (cf. [4, Section 1.3]). Lemma 5.1. For any two complexes A • and B • ∈ C ( E ) , the subgroup in Ext C ( E ) ( B • , A • ) formed by the termwise split short exact sequences is naturallyisomorphic to the group Hom K ( E ) ( B • , A • [1]) . In particular, if either (a) A • is a complex of injective objects in E , or (b) B • is a complex of projective objects in E ,then Ext C ( E ) ( B • , A • ) ≃ Hom K ( E ) ( B • , A • [1]) .Proof. The first assertion can be formulated using the termwise split exact categorystructure on C ( E ); then it holds for any additive category E . The natural mapHom K ( E ) ( B • , A • [1]) −→ Ext C ( E ) ( B • , A • ) takes the homotopy class of a morphism f : B • −→ A • [1] to the equivalence class of the termwise split short exact sequence0 −→ A • −→ cone( f )[ − −→ B • −→ 0, where cone( f ) denotes the cone of amorphism of complexes. We leave further (elementary) details to the reader. (cid:3) Denote by E gr = Q i ∈ Z E the additive category of graded objects in an additivecategory E . Following [35] and [4], we denote the functor of forgetting the differential C ( E ) −→ E gr by C • C • ♯ .The functor ( − ) ♯ has adjoints on both sides; we denote the left adjoint functorto ( − ) ♯ by G + : E gr −→ C ( E ) and the right adjoint one by G − : E gr −→ C ( E ). Thefunctors G + and G − were originally introduced for CDG-modules in [35, proof ofTheorem 3.6]; in the case of complexes in an additive category, they are much eas-ier to describe. To a graded object E = ( E i ) i ∈ Z ∈ E gr , the functor G + assignsthe contractible complex with the terms G + ( E ) i = E i − ⊕ E i and the differential d i : G + ( E ) i −→ G + ( E ) i +1 given by the 2 × E i −→ E i . The complex G − ( E ) has theterms G − ( E ) i = E i ⊕ E i +1 ; otherwise it is constructed similarly to G + ( E ). So thetwo functors only differ by the shift: one has G − = G + [1].Given an abelian category B , we denote by B grproj the category of projective objectsin the abelian category B gr or, which is the same, the category of graded objectsin the additive category B proj . Furthermore, we denote by C ( B proj ) the category ofcomplexes of projective objects in B and by C ( B ) proj the category of projective objectsin the abelian category C ( B ). Similar notation is used for injective objects/gradedobjects/complexes in an abelian category A .The abelian category C ( E ) has enough projective (respectively, injective) objectswhenever an abelian category E has. The next lemma describes such projective orinjective complexes. emma 5.2. (a) For any abelian category A with enough injective objects, a complex J • ∈ C ( A ) is an injective object of C ( A ) if and only if J • is contractible and all itscomponents J i are injective objects of C ( A ) , and if and only if J • has the form G − ( I ) for some collection of injective objects ( I i ) i ∈ Z in A . Symbolically, J • ∈ C ( A ) inj if andonly if J • is contractible and J • ∈ C ( A inj ) , and if and only if J • ∈ G − ( A grinj ) . (b) For any abelian category B with enough projective objects, a complex P • ∈ C ( B ) is an projective object of C ( B ) if and only if P • is contractible and all its components P i are projective objects of C ( B ) , and if and only if P • has the form G + ( Q ) for somecollection of projective objects ( Q i ) i ∈ Z in B . Symbolically, P • ∈ C ( B ) proj if and onlyif P • is contractible and P • ∈ C ( B proj ) , and if and only if P • ∈ G + ( B grproj ) .Proof. This is our version of [4, Lemma 1.3.3]. We leave the details to the reader. (cid:3) Lemma 5.3. (a) Let A be an abelian category with enough injectives, and let ( W , R ) be a cotorsion pair in C ( A ) such that R ⊂ C ( A inj ) . Assume that the cotorsion pair ( W , R ) is invariant under the shift: W = W [1] , or equivalently, R = R [1] . Then R ∩ W = C ( A ) inj . If, moreover, ( W , R ) is complete, then W is thick in C ( A ) . (b) Let B be an abelian category with enough projectives, and let ( L , W ) be a co-torsion pair in C ( B ) such that L ⊂ C ( B proj ) . Assume that the cotorsion pair ( L , W ) isinvariant under the shift: W = W [1] , or equivalently, L = L [1] . Then L ∩ W = C ( B ) proj .If, moreover, ( L , W ) is complete, then W is thick in C ( A ) .Proof. This is our version of [4, Lemma 1.3.4], and we follow the argument in [4].Part (a): the inclusion C ( A ) inj ⊂ W ⊥ = R is obvious. The inclusion C ( A ) inj ⊂ ⊥ C ( A inj ) holds, because C ( A ) inj is the category of projective-injective objects of theFrobenius exact category C ( A inj ). Specifically, for any J • ∈ C ( A ) inj and I • ∈ C ( A inj )we have, by Lemmas 5.1(a) and 5.2(a), Ext C ( A ) ( J • , I • ) ≃ Hom K ( A ) ( J • , I • [1]) = 0 sincethe complex J • is contractible. Hence C ( A ) inj ⊂ ⊥ R = W . To prove the inclusion R ∩ W ⊂ C ( A ) inj , it suffices to show that every complex J • ∈ R ∩ W is contractible(in view of Lemma 5.2(a), as we already know that R ∩ W ⊂ R ⊂ C ( A inj )). Indeed,by Lemma 5.1(a), Hom K ( A ) ( J • , J • ) = Ext C ( A ) ( J • , J • [ − R is closed under cosyzygies in C ( A )(each J • ∈ R admits a short exact sequence 0 −→ J • −→ G − ( J • ♯ ) −→ J • [1] −→ G − ( J • ♯ ) ∈ C ( A ) inj by the previous lemma). It follows by a standard dimensionshifting argument that the cotorsion pair ( W , R ) is hereditary. Now we just applyLemma 4.6(a).Part (b) is dual. (cid:3) Locally Presentable Abelian Categorieswith a Projective Generator In this section we study the conventional derived category D ( B ) of a locally pre-sentable abelian category B with enough projective objects. We start with the fol-lowing lemma describing the class of abelian categories we are interested in. emma 6.1. A locally presentable abelian category B has enough projective objectsif and only if it has a single projective generator.Proof. The “if” implication holds, since any abelian category with coproducts and aprojective generator has enough projective objects. To prove the “only if”, observethat any abelian category with enough projective objects and a set of generators hasa set of projective generators, and any abelian category with coproducts and a set ofprojective generators has a single projective generator. (cid:3) Lemma 6.2. For any abelian category B with coproducts and a set of projectivegenerators { P α } , one has Fil ( { P α } ) ⊕ = Add ( { P α } ) = B proj .Proof. The assertion is straightforward. The least obvious part is the inclusion Fil ( B proj ) ⊂ B proj , which can be also deduced from Proposition 1.3. (cid:3) The following lemma is quite general. Lemma 6.3. For any locally presentable additive category E and any small additivecategory X , the additive category AdF ( X , E ) of additive functors X −→ E is locallypresentable. In particular, the category of complexes C ( E ) is locally presentable.Proof. Let X be the set of objects of the small category X . Consider the product E X of X copies of E . Then the forgetful functor AdF ( X , E ) −→ E X is monadic, i. e.,it identifies AdF ( X , E ) with the Eilenberg–Moore category of a monad on E X . Thisassertion holds for any cocomplete additive category E . Moreover, the underlyingfunctor of the relevant monad on E X preserves all colimits. Now if E is locallypresentable, then E X is locally presentable by [1, Proposition 2.67], and it followsthat AdF ( X , E ) is locally presentable by [1, Theorem and Remark 2.78]. It remains tomention that the category of complexes C ( E ) is isomorphic to AdF ( X , E ) for a suitablechoice of X , see [47, Proposition 4.5].Here is also a direct elementary argument showing that C ( E ) is locally presentable.Clearly, the category C ( E ) is cocomplete whenever the category E is. Given an object E ∈ E , consider E as a one-term complex concentrated in degree 0. Then thefunctor E G + ( E )[ − i ] : E −→ C ( E ) is left adjoint to the functor C ( E ) −→ E taking a complex C • to its degree- i term C i . Assume that the category E is locally λ -presentable for some regular cardinal λ , and let S be a (strongly) generating setof λ -presentable objects in E . Then the objects G + ( S )[ − i ], S ∈ S , i ∈ Z form a(strongly) generating set of λ -presentable objects in C ( E ). Hence the category C ( E )is locally λ -presentable by [1, Theorem 1.20]. (cid:3) Let B be an abelian category. A complex P • ∈ K ( B ) is said to be homotopyprojective if Hom K ( B ) ( P • , X • ) = 0 for any acyclic complex X • in B . We denote thefull subcategory of homotopy projective complexes by K ( B ) hpr ⊂ K ( B ). Furthermore,let us denote by K ( B proj ) hpr = K ( B proj ) ∩ K ( B ) hpr ⊂ K ( B ) the full subcategory of homotopy projective complexes of projective objects in the homotopy category K ( B ).Clearly, both K ( B ) hpr and K ( B proj ) hpr are triangulated subcategories in K ( B ). roposition 6.4. For any abelian category B , the composition of the fully faithfulinclusion K ( B ) hpr −→ K ( B ) with the Verdier quotient functor K ( B ) −→ D ( B ) is a fullyfaithful functor K ( B ) hpr −→ D ( B ) . Hence we have a pair of fully faithful triangulatedfunctors K ( B proj ) hpr −→ K ( B ) hpr −→ K ( B ) .Proof. More generally, for any triangulated category H and full triangulated subcat-egories P and X ⊂ H such that Hom H ( P, X ) = 0 for all P ∈ P and X ∈ X , thecomposition of the inclusion P −→ H with the Verdier quotient functor H −→ H / X is fully faithful. In the situation at hand, take H = K ( B ) and P = K ( B ) hpr , and let X = K ( B ) ac ⊂ K ( B ) be the full subcategory of acyclic complexes. (cid:3) We will say that an abelian category B has enough homotopy projective complexes if the functor K ( B ) hpr −→ D ( B ) is a triangulated equivalence. Moreover, we will saythat B has enough homotopy projective complexes of projective objects if the functor K ( B proj ) hpr −→ D ( B ) is a triangulated equivalence.In the latter case, clearly, the inclusion K ( B proj ) hpr −→ K ( B ) hpr is a triangulatedequivalence, too. So, in an abelian category with enough homotopy projective com-plexes of projective objects, every homotopy projective complex is homotopy equiv-alent to a homotopy projective complex of projective objects.Introduce the notation C ( B ) ac ⊂ C ( B ) for the full subcategory of acyclic complexesin the category of complexes C ( B ) and the notation C ( B proj ) hpr ⊂ C ( B ) for the fullsubcategory of homotopy projective complexes of projective objects in C ( B ). That is,these categories are the full preimages of K ( B ) ac and K ( B proj ) hpr , respectively, underthe natural functor C ( B ) −→ K ( B ). Theorem 6.5. Let B be a locally presentable abelian category with enough projectiveobjects. Then the pair of classes of objects C ( B proj ) hpr and C ( B ) ac is a hereditarycomplete cotorsion pair in the abelian category C ( B ) .Proof. Let P be a projective generator of B . The claim is that the cotorsion pair weare interested in is generated by the set of objects S = { P [ i ] } i ∈ Z ⊂ C ( B ) (where P isviewed as a one-term complex concentrated in degree 0).Indeed, for any complex C • ∈ C ( B ) we have, by Lemma 5.1(b), Ext C ( B ) ( P [ i ] , C • ) ≃ Hom K ( B ) ( P [ i ] , C • [1]) ≃ Hom B ( P, H − i ( C • )). The latter abelian group vanishes if andonly if H − i ( C • ) = 0. Hence S ⊥ = C ( B ) ac ⊂ C ( B ).In the pair of adjoint functors ( − ) ♯ : C ( B ) −→ B gr and G − : B gr −→ C ( B ), boththe functors are exact. Hence for any complex C • ∈ C ( B ) and any graded object B ∈ B gr we have Ext n C ( B ) ( C • , G − ( B )) ≃ Ext n B gr ( C • ♯ , B ) for all n ≥ 0. In particular,this isomorphism holds for n = 1. Since the complex G − ( B ) is acyclic, we haveshown that ⊥ ( C ( B ) ac ) ⊂ C ( B proj ). Now for a complex Q • ∈ C ( B proj ) and any complex X • ∈ C ( B ) we have Ext C ( B ) ( Q • , X • ) ≃ Hom K ( B ) ( Q • , X • [1]) by Lemma 5.1(b). Thus ⊥ ( C ( B ) ac ) = C ( B proj ) hpr ⊂ C ( B ). So our pair of classes of objects is indeed thecotorsion pair generated by the set S ⊂ C ( B ).Obviously, any complex C • is a subcomplex of an acyclic (or even contractible)complex via the adjunction morphism C • −→ G − ( C • ♯ ); so the class C ( B ) ac is cogen-erating in C ( B ). It is also easy to present any complex in B as a quotient complex f a contractible complex of projective objects (since there are enough projectivesin B ). All contractible complexes are obviously homotopy projective. Hence the class C ( B proj ) hpr is generating in C ( B ).By Lemma 6.3, the category C ( B ) is locally presentable. Applying Theorem 3.3,we conclude that our cotorsion pair is complete. The cotorsion pair is hereditary,since the class C ( B ) ac is closed under the cokernels of monomorphisms in C ( B ). (cid:3) Corollary 6.6. Any locally presentable abelian category with a projective generatorhas enough homotopy projective complexes of projective objects. In other words, thenatural functor K ( B proj ) hpr −→ D ( B ) is a triangulated equivalence.Proof. Let B be a locally presentable abelian category with enough projective objects.Given a complex C • ∈ C ( B ), we need to find a homotopy projective complex ofprojective objects Q • together with a quasi-isomorphism Q • −→ C • of complexesin B . For this purpose, consider a special precover short exact sequence 0 −→ X • −→ Q • −→ C • −→ X • ∈ C ( B ) ac and Q • ∈ C ( B proj ) hpr . Since the complex X • is acyclic, it follows that theepimorphism of complexes Q • −→ C • is a quasi-isomorphism. (cid:3) The proof of Theorem 6.5 also provides us with the following description of theclass of homotopy projective complexes of projectives. Corollary 6.7. Let B be a locally presentable abelian category with enough projectiveobjects. If we choose a projective generator P and denote S = { P [ i ] } i ∈ Z ⊂ C ( B ) , then C ( B proj ) hpr = Fil ( S ) ⊕ ⊂ C ( B ) .Proof. We know that C ( B proj ) hpr is the left hand side of the cotorsion pair in C ( B )generated by S . Moreover, Fil ( S ) is generating since the objects G + ( P )[ i ], i ∈ Z ,form a set of projective generators of C ( B ) and, obviously, G + ( P )[ i ] ∈ Fil ( S ). UsingLemma 6.2, one can see that the class Fil ( S ) is generating (notice that Fil ( Fil ( S )) = Fil ( S ) by [41, Lemma 4.6(d)]), and we just apply Theorem 3.4. (cid:3) Theorem 6.8. Let B be a locally presentable abelian category with enough projectiveobjects. Then the triple of classes of objects L = C ( B proj ) hpr , W = C ( B ) ac , and R = C ( B ) is a cofibrantly generated hereditary abelian model structure on the abeliancategory of complexes C ( B ) .Proof. The pair of classes ( L , W ) is a complete cotorsion pair in C ( B ) by Theorem 6.5.The abelian category of complexes C ( B ) has enough projective objects, since theabelian category B has. According to Lemma 5.3(b), it follows that L ∩ W = C ( B ) proj .The class of acyclic complexes W is obviously thick in C ( B ) (see also Lemma 4.6(b)or 5.3(b)). By Lemma 4.5(b) (applied to the category C ( B )), the triple ( L , W , R ) is aprojective abelian model structure on C ( B ).As explained in Section 4, any projective abelian model structure is hereditary.Finally, it is clear from the proof of Theorem 6.5 that the cotorsion pair ( L , W ) in C ( B )is generated by a set of objects. The cotorsion pair ( C ( B ) proj , C ( B )) is generated by theempty set of objects (or by the single zero object) in C ( B ). According to Corollary 4.3,this means that our abelian model structure is cofibrantly generated. (cid:3) he abelian model structure ( L , W , R ) defined in Theorem 6.8 is called the projectivederived model structure on the abelian category of complexes C ( B ). Lemma 6.9. For any locally presentable abelian category B with enough projectiveobjects, the class W of all weak equivalences in the projective derived model structureon the abelian category C ( B ) coincides with the class of all quasi-isomorphisms ofcomplexes in B .Proof. According to [22, Lemma 5.8], a monomorphism in an abelian model cate-gory is a weak equivalence if and only if its cokernel is weakly trivial. Dually, anepimorphism is a weak equivalence if and only if its kernel is weakly trivial.In the situation at hand, the class of weakly trivial objects W is the class of allacyclic complexes. This implies the assertion of the lemma for all monomorphismsand epimorphisms of complexes. It remains to recall that any morphism in C ( B )is the composition of, say, a trivial cofibration (which is a monomorphism, a weakequivalence, and a quasi-isomorphism) and a fibration (which is an epimorphism),and that both the classes of weak equivalences and quasi-isomorphisms satisfy thetwo-out-of-three property. Alternatively, one can use the fact that any morphism isthe composition of a cofibration and a trivial fibration. (cid:3) The following corollary presumes existence of (set-indexed) coproducts in the de-rived category D ( B ). Notice that such coproducts can be simply constructed as thecoproducts in the homotopy category K ( B ) hpr or K ( B proj ) hpr , which is equivalent to D ( B ) by Corollary 6.6. The full subcategory K ( B ) hpr is closed under coproducts in K ( B ), so the coproducts in K ( B ) hpr (just as in K ( B )) can be computed as the termwisecoproducts of complexes. Corollary 6.10. For any locally presentable abelian category B with enough projec-tive objects, the (unbounded) derived category D ( B ) is a well-generated triangulatedcategory (in the sense of the book [31] and the paper [25] ).Proof. Following [4, Corollary 1.1.15 and the preceding discussion], any hereditaryabelian model category is stable (so its homotopy category is triangulated). Hence,the derived category D ( B ) can be equivalently defined as the homotopy category C ( B )[ W − ] of the stable combinatorial model category C ( B ) with the projective de-rived model category structure. The conclusion follows by [44, Proposition 6.10]or [11, Theorems 3.1 and 3.9]. (cid:3) The next result appeared in the context of DG-contramodules over a DG-coalgebra(over a field) in [35, Section 5.5]; see [36, Theorem 1.1(b)]. Theorem 6.11. Let B be a locally presentable abelian category with a projective gen-erator P . Then the category D ( B ) is generated, as a triangulated category with coprod-ucts, by the single object P (viewed as a one-term complex concentrated in degree ).In other words, the full subcategory of homotopy projective complexes K ( B ) hpr ⊂ K ( B ) is the minimal strictly full triangulated subcategory in K ( B ) containing the object P and closed under coproducts. roof. The key observation is that Hom D ( B ) ( P, X • [ i ]) = 0 for all i ∈ Z implies X • = 0for a given object X • ∈ D ( B ), since for any complex X • one has a natural isomorphismof abelian groups Hom D ( B ) ( P, X • [ i ]) ≃ Hom B ( P, H i ( X • )).By Corollary 6.10, the triangulated category D ( B ) is well-generated. Denote by D ′ ⊂ D ( B ) the minimal full triangulated subcategory in D ( B ) containing P and closedunder coproducts. By [27, Theorem 7.2.1(2)], the localizing subcategory generated byany set of objects in a well-generated triangulated category is also well-generated; so D ′ is well-generated. Any well-generated triangulated category is perfectly generatedby definition; so the Brown representability theorem [27, Theorem 5.1.1] is applicable,and in particular the inclusion functor D ′ −→ D ( B ) has a right adjoint.The right adjoint functor to a fully faithful triangulated functor is a Verdier quo-tient functor. The kernel of our functor D ( B ) −→ D ′ consists of complexes X • satisfying Hom D ( B ) ( P [ i ] , X • ) = 0 for all i ∈ Z , since P [ i ] ∈ D ′ . Thus all such objects X • ∈ D ( B ) vanish, hence the functor D ( B ) −→ D ′ is a triangulated equivalence; andit follows that the inclusion D ′ −→ D ( B ) is a triangulated equivalence, too; the lattermeans that D ′ = D ( B ), as desired. (cid:3) Contraderived Model Structure In this section we consider contraderived categories in the sense of Becker [4].In well-behaved cases, this means simply the homotopy category of complexes ofprojective objects (which was studied first by Jørgensen [23] in the case of modulecategories); see the discussion in the introduction.The contraderived category in the sense of Becker needs to be distinguished fromthe contraderived category in the sense of the books and papers [34, 35, 37, 43, 38].The two definitions of a contraderived category are known to be equivalent undercertain assumptions [35, Theorem 3.8], but it is still an open question whether theyare equivalent for the category of modules over an arbitrary associative ring (see [38,Example 2.6(3)] for a discussion).Let B be an abelian category. A complex X • ∈ K ( B ) is said to be contraacyclic (in the sense of Becker) if Hom K ( B ) ( P • , X • ) = 0 for any complex of projective ob-jects P • ∈ K ( B proj ). We denote the full subcategory of contraacyclic complexes by K ( B ) ctrac ⊂ K ( B ). Clearly, K ( B ) ctrac is a triangulated (and even thick) subcategory inthe homotopy category K ( B ). The quotient category D ctr ( B ) = K ( B ) / K ( B ) ctrac is calledthe contraderived category of B (in the sense of Becker). Lemma 7.1. (a) For any short exact sequence −→ K • −→ L • −→ M • −→ ofcomplexes in B , the total complex Tot( K • → L • → M • ) of the bicomplex with threerows K • −→ L • −→ M • belongs to K ( B ) ctrac . (b) The full subcategory of contraacyclic complexes K ( B ) ctrac is closed under productsin the homotopy category K ( B ) .Proof. This is our version of [35, Theorem 3.5(b)]. Part (b) follows immediately fromthe definitions. To prove part (a), let us introduce the following notation. or any two complexes C • and D • in an additive category E , let Hom E ( C • , D • )denote the complex of morphisms from C • to D • . So the degree n componentHom n E ( C • , D • ) of the complex Hom E ( C • , D • ) is the group Hom E gr ( C • ♯ , D • ♯ [ n ]) ofmorphisms C • ♯ −→ D • ♯ [ n ] in the category E gr of graded objects in E . The groupHom C ( E ) ( C • , D • ) can be computed as the kernel of the differential Hom E ( C • , D • ) −→ Hom E ( C • , D • ), while the group Hom K ( E ) ( C • , D • ) is the degree 0 cohomology group H (Hom E ( C • , D • )) of the complex Hom E ( C • , D • ).Now in the situation at hand, for any complex of projective objects P • ∈ K ( B proj )and any short exact sequence of complexes 0 −→ K • −→ L • −→ M • −→ B wehave a short exact sequence of complexes of abelian groups 0 −→ Hom B ( P • , K • ) −→ Hom B ( P • , L • ) −→ Hom B ( P • , M • ) −→ 0. The complex Hom B ( P • , Tot( K • → L • → M • )) can be computed as the total complex of the bicomplex of abelian groupsHom B ( P • , K • ) −→ Hom B ( P • , L • ) −→ Hom B ( P • , M • ). It remains to observe that thetotalization of any short exact sequence of abelian groups is an acyclic complex. (cid:3) Proposition 7.2. Let B be a locally presentable abelian category with enough pro-jective objects. Then there exists a set of complexes of projective objects S ⊂ C ( B proj ) such that the class of all complexes of projective objects C ( B proj ) ⊂ C ( B ) is the classof all direct summands of complexes filtered by S , that is C ( B proj ) = Fil ( S ) ⊕ .First proof. The stronger statement that there exists a set S ′ ⊂ C ( B proj ) such that C ( B proj ) = Fil ( S ′ ) is provable using the results of [29, Section A.1.5], [30, Section 3],and [41, Lemma 4.6]. Without going into details, let us describe the overall logicof this argument, generalizing [30, Remark 3.5], while a direct and more elementaryargument is presented in the second proof below. The idea is to assign to everyclass of objects L in an abelian category a related class of morphisms, namely theclass of all L -monomorphisms L - M ono , and use the known results about cofibrantlygenerated weak factorization systems in locally presentable categories.We will say that a class of objects L in an abelian category E is transmonic if anytransfinite composition of L -monomorphisms is a monomorphism. Then it is claimedthat, for any transmonic set of objects S in a locally presentable abelian category E , there exists a set of objects S ′ ⊂ E such that Fil ( S ′ ) = Fil ( S ) ⊕ . This essentiallyfollows from [29, Proposition A.1.5.12] in view of Proposition 2.5. Furthermore, aclass of objects L ⊂ E is said to be deconstructible if there exists a set S ⊂ E suchthat L = Fil ( S ). In other words, if L is transmonic and deconstructible in E , so is L ⊕ .For any small additive category X and any transmonic deconstructible class ofobjects L in a locally presentable abelian category E , the class AdF ( X , L ) of all additivefunctors X −→ L ( L being viewed as a full subcategory in E ) is (transmonic and)deconstructible in the locally presentable abelian category AdF ( X , E ) of all additivefunctors X −→ E . This is essentially [30, Corollary 3.4] or a particular case of [30,Corollary 3.6], in view of Proposition 2.5.In particular, for any locally presentable abelian category B with a projective gen-erator P , we start with the transmonic and deconstructible class Fil ( { P } ) consistingof all coproducts of copies of P . As this class is transmonic and deconstructible, sois the class of all projective objects B proj = Fil ( { P } ) ⊕ , and it further follows, for a uitable choice of the additive category X (see the proof of [47, Proposition 4.5]), thatthe class C ( B proj ) is deconstructible in C ( B ). (cid:3) Second proof. Here is a direct proof of the assertion stated in Proposition 7.2. It isclear from Lemma 6.2 that Fil ( C ( B proj )) ⊕ ⊂ C ( B proj ) (as the forgetful functor C ( B ) −→ B gr preserves extensions and colimits, hence it also preserves transfinitely iteratedextensions in the sense of the directed colimit). So it suffices to find a set of objects S ⊂ C ( B proj ) such that C ( B proj ) ⊂ Fil ( S ) ⊕ . We will actually find such a set of boundedbelow complexes, following the argument for [47, Proposition 4.9, (1) = ⇒ (2)].Let P be a projective generator of B . Then any complex of projective objectsin B is a direct summand of a complex whose terms are coproducts of copies of P .Choose an uncountable regular cardinal κ such that the object P ∈ B is κ -presentable(then the category B is locally κ -presentable). Let S be the set of (representativesof isomorphism classes) of bounded below complexes whose terms are coproducts ofless than κ copies of P . We claim that any complex in B whose terms are coproductsof copies of P belongs to Fil ( S ) ⊂ C ( B ).Let Q • be a complex in B whose term Q n = P ( X n ) is the coproduct of copies of P indexed by a set X n , for every n ∈ Z . Let α be the successor cardinal of thecardinality of the disjoint union ` n ∈ Z X n . Proceeding by transfinite induction onordinals 0 ≤ β ≤ α , we will construct a smooth chain of subsets Y nβ ⊂ X n suchthat Y n = ∅ and Y nα = X n for every n ∈ Z , the cardinality of Y nβ +1 \ Y nβ is smallerthan κ for every 0 ≤ β < α and n ∈ Z and is empty for n ≪ 0, and the gradedsubobject with the terms Q nβ = P ( Y nβ ) ⊂ P ( X n ) = Q n is a subcomplex Q • β in Q • forevery 0 ≤ β ≤ α .For every element x ∈ X n , let ι x : P −→ P ( X n ) = Q n be the direct summandinclusion corresponding to the element x . Since the object P is κ -presentable, thereexists a subset Z x ⊂ X n +1 of the cardinality less than κ such that the compositionof ι x with the differential d n : Q n −→ Q n +1 factorizes through the direct summand(subcoproduct) inclusion P ( Z x ) −→ P ( X n +1 ) = Q n +1 .Suppose that the subsets Y nγ ⊂ X n have been constructed already for all γ < β and n ∈ Z . For a limit ordinal β , we put Y nβ = S γ<β Y nγ for every n ∈ Z . For asuccessor ordinal β = γ + 1, if Y nγ = X n for every n ∈ Z , then we put Y nβ = X n aswell. Otherwise, choose m ∈ Z and an element z ∈ X m \ Y mγ .Proceeding by induction on n ≥ m , define subsets Z n ⊂ X n by the rules Z m = { z } and Z n +1 = S x ∈ Z n Z x . Then · · · −→ −→ P = P ( Z m ) −→ P ( Z m +1 ) −→ P ( Z m +2 ) −→· · · is a subcomplex in Q • , and the cardinality of the set Z n is smaller than κ for every n ≥ m . It remains to put Y nβ = Y nγ for n < m and Y nβ = Y nγ ∪ Z n for n ≥ m . (cid:3) Introduce the notation C ( B ) ctrac for the full subcategory of contraacyclic complexesin C ( B ). So C ( B ) ctrac ⊂ C ( B ) is the full preimage of K ( B ) ctrac ⊂ K ( B ) under the naturalfunctor C ( B ) −→ K ( B ). Theorem 7.3. Let B be a locally presentable abelian category with enough projec-tive objects. Then the pair of classes of objects C ( B proj ) and C ( B ) ctrac is a hereditarycomplete cotorsion pair in the abelian category C ( B ) . roof. Notice that the abelian category C ( B ) is locally presentable by Lemma 6.3.Let S be a set of complexes of projective objects in B such that C ( B proj ) = Fil ( S ) ⊕ , asin Proposition 7.2. The claim is that the set S ⊂ C ( B ) generates the cotorsion pairwe are interested in.Indeed, for any complexes Q • ∈ C ( B proj ) and C • ∈ C ( B ) we have Ext C ( B ) ( Q • , C • ) ≃ Hom K ( B ) ( Q • , C • [1]) by Lemma 5.1(b). Hence C ( B proj ) ⊥ = C ( B ) ctrac ⊂ C ( B ). By Propo-sition 1.3, it follows that S ⊥ = C ( B ) ctrac .Furthermore, the class C ( B proj ) = Fil ( S ) ⊕ is clearly generating in C ( B ). Apply-ing Theorem 3.4, we can conclude that C ( B proj ) = ⊥ ( C ( B ) ctrac ). Alternatively, onecan argue as in the proof of Theorem 6.5 in order to show that any complex leftExt -orthogonal to all contractible complexes in C ( B ) is a complex of projective ob-jects (as the complexes G − ( B ) are contractible).Any complex is a subcomplex of a contractible complex, so the class C ( B ) ctrac is co-generating in C ( B ). Hence Theorem 3.3 is applicable, and we can conclude thatour cotorsion pair is complete. The cotorsion pair is hereditary, since the class C ( B ) ctrac is closed under the cokernels of monomorphisms in C ( B ), as one can seefrom Lemma 7.1(a). It is also clear that the class C ( B proj ) ⊂ C ( B ) is closed under thekernels of epimorphisms. (cid:3) Corollary 7.4. For any locally presentable abelian category B with enough projectiveobjects, the composition of the inclusion of triangulated categories K ( B proj ) −→ K ( B ) with the Verdier quotient functor K ( B ) −→ D ctr ( B ) is a triangulated equivalence K ( B proj ) ≃ D ctr ( B ) .Proof. It is clear from the definitions that the triangulated functor K ( B proj ) −→ D ctr ( B ) is fully faithful (cf. the proof of Proposition 6.4). In order to prove thecorollary, it remains to find, for any complex C • ∈ K ( B ), a complex of projectiveobjects Q • ∈ K ( B proj ) together with a morphism of complexes Q • −→ C • whose conebelongs to K ( B ) ctrac .For this purpose, consider a special precover short exact sequence 0 −→ X • −→ Q • −→ C • −→ X • ∈ C ( B ) ctrac and Q • ∈ C ( B proj ). By Lemma 7.1(a), the totalization Tot( X • → Q • → C • ) ofthe short exact sequence 0 −→ X • −→ Q • −→ C • −→ X • is contraacyclic, it follows that the cone of the morphism ofcomplexes Q • −→ C • is contraacyclic, too. (cid:3) In terms of the full subcategory K ( B proj ) ⊂ K ( B ), Corollary 7.4 means that thetriangulated inclusion functor K ( B proj ) −→ K ( B ) has a right adjoint. We also referto [8, Theorem 3.5] for another argument to deduce this from Theorem 7.3. Theorem 7.5. Let B be a locally presentable abelian category with enough projectiveobjects. Then the triple of classes of objects L = C ( B proj ) , W = C ( B ) ctrac , and R = C ( B ) is a cofibrantly generated hereditary abelian model structure on the abelian categoryof complexes C ( B ) .Proof. This is similar to the proof of Theorem 6.8. The pair of classes ( L , W )form a hereditary complete cotorsion pair in C ( B ) by Theorem 7.3. According to emma 5.3(b), it follows that L ∩ W = C ( B ) proj and that the class of contraacycliccomplexes W is thick in C ( B ) (see also Lemma 7.1(a)). By Lemma 4.5(b), the triple( L , W , R ) is a projective abelian model structure on the category C ( B ). Finally, it wasshown in the proof of Theorem 7.3 that the cotorsion pair ( L , W ) in C ( B ) is generatedby a set of objects. For the cotorsion pair ( C ( B ) proj , C ( B )), the same was explainedin the proof of Theorem 6.8. (cid:3) The abelian model structure ( L , W , R ) defined in Theorem 7.5 is called the con-traderived model structure on the abelian category of complexes C ( B ). Lemma 7.6. For any locally presentable abelian category B with enough projectiveobjects, the class W of all weak equivalences in the contraderived model structure onthe abelian category C ( B ) coincides with the class of all morphisms of complexes in B with the cones belonging to C ( B ) ctrac .Proof. Similar to the proof of Lemma 6.9. One needs to notice that, by Lemma 7.1(a),a monomorphism of complexes in C ( B ) has contraacyclic cokernel if and only if it hascontraacyclic cone, and similarly, an epimorphism of complexes in C ( B ) has contra-acyclic kernel if and only if it has contraacyclic cone. (cid:3) The following corollary presumes existence of (set-indexed) coproducts in the con-traderived category D ctr ( B ). Notice that such coproducts can be simply constructedas the coproducts in the homotopy category K ( B proj ), which is equivalent to D ctr ( B )by Corollary 7.4. Corollary 7.7. For any locally presentable abelian category B with enough projectiveobjects, the contraderived category D ctr ( B ) is a well-generated triangulated category(in the sense of the book [31] and the paper [25] ).Proof. Similar to the proof of Corollary 6.10. One only needs to notice that thecontraderived category D ctr ( B ) can be equivalently defined as the homotopy category C ( B )[ W − ] of the contraderived model structure on C ( B ). The key observation isthat, for any additive category E , inverting all the homotopy equivalences in thecategory of complexes C ( E ) produces the homotopy category K ( E ) (i. e., homotopicmorphisms become equal after inverting the homotopy equivalences); see [17, III.4.2–3] (cf. [34, last paragraph of the proof of Theorem 2.5]). Then in the situation athand, it follows that inverting all the morphisms with contraacyclic cones in C ( B )produces the contraderived category D ctr ( B ). (cid:3) Proposition 7.8. Let E be an accessible additive category with coproducts and M ∈ E be an object. Then there exists a (unique) locally presentable abelian category B withenough projective objects such that the full subcategory B proj ⊂ B is equivalent to thefull subcategory Add E ( M ) ⊂ E .Proof. Any accessible additive category is idempotent-complete [1, Observation 2.4].For any idempotent-complete additive category E with coproducts and any object M ∈ E , there exists a unique abelian category B with enough projective objectssuch that the full subcategory of projective objects B proj ⊂ B is equivalent to the ull subcategory Add E ( M ) ⊂ E [43, Theorem 1.1(a)], [39, Examples 1.2(1–2)]. Thecategory B is locally presentable if and only if the object M ∈ E is abstractly κ -smallfor some cardinal κ ; and any κ -presentable object in E is abstractly κ -small [41,Section 1.1], [42, Section 6.4 and Proposition 9.1], [39, Section 1]. (cid:3) Now we can elegantly recover a consequence of [47, Proposition 4.9]. Corollary 7.9. For any accessible additive category E with coproducts and any object M ∈ E , the homotopy category K ( Add E ( M )) of (unbounded complexes in) the additivecategory Add E ( M ) ⊂ E is a well-generated triangulated category.Proof. By Proposition 7.8, there exists a locally presentable abelian category B withenough projective objects such that the additive category Add E ( M ) is equivalentto B proj . By Corollary 7.4, the homotopy category K ( Add E ( M )) ≃ K ( B proj ) can beinterpreted as the contraderived category D ctr ( B ) (in the sense of Becker). Accordingto Corollary 7.7, the triangulated category D ctr ( B ) is well-generated. (cid:3) Grothendieck Abelian Categories In this section we discuss the conventional derived category D ( A ) of a Grothendieckabelian category A . Recall that a cocomplete abelian category is said to be Grothendieck if it has exact functors of directed colimit and a set of generators.The abelian category of complexes C ( A ) in a Grothendieck abelian category A (or more generally, any category of additive functors AdF ( X , A ), as in Lemma 6.3)is again a Grothendieck abelian category. It is well-known that all Grothendieckcategories are locally presentable (for a reference, see [28, Corollary 5.2], and for ageneralization, [41, Theorem 2.2]).Let A be an abelian category. A complex J • ∈ K ( A ) is said to be homotopyinjective if Hom K ( A ) ( X • , J • ) = 0 for any acyclic complex X • in A . We denote the fullsubcategory of homotopy injective complexes by K ( A ) hin ⊂ K ( A ). Furthermore, letus denote by K ( A inj ) hin = K ( A inj ) ∩ K ( A ) hin ⊂ K ( A ) the full subcategory of homotopyinjective complexes of injective objects in the homotopy category K ( A ). Both K ( A ) hin and K ( A inj ) hin are triangulated subcategories in K ( A ). Proposition 8.1. For any abelian category A , the composition of the fully faithfulinclusion K ( A ) hin −→ K ( A ) with the Verdier quotient functor K ( A ) −→ D ( A ) is a fullyfaithful functor K ( A ) hin −→ D ( A ) . Hence we have a pair of fully faithful triangulatedfunctors K ( A inj ) hin −→ K ( A ) hin −→ K ( A ) .Proof. This is a dual assertion to Proposition 6.4. (cid:3) We will say that an abelian category A has enough homotopy injective complexes if the functor K ( A ) hin −→ D ( A ) is a triangulated equivalence. Moreover, we will saythat A has enough homotopy injective complexes of injective objects if the functor K ( A inj ) hin −→ D ( A ) is a triangulated equivalence. n the latter case, clearly, the inclusion K ( A inj ) hin −→ K ( A ) hin is a triangulatedequivalence, too. So, in an abelian category with enough homotopy injective com-plexes of injective objects, any homotopy injective complex is homotopy equivalentto a homotopy injective complex of injective objects.As in Section 6, we denote by C ( A ) ac ⊂ C ( A ) the full subcategory of acyclic com-plexes in C ( A ). We also denote by C ( A inj ) hin ⊂ C ( A ) the full subcategory of homotopyinjective complexes of injective objects in C ( A ). Lemma 8.2. Let A be a Grothendieck abelian category. Then there exists a set S of objects in A such that all the objects of A are filtered by objects from S , that is A = Fil ( S ) .Proof. This is [4, Example 1.2.6(1)]. Choose a set of generators S for the abelian cat-egory A , and let S consist of all (representatives of isomorphism classes) of quotientobjects of the objects from S in A . Let M ∈ A be an object. Consider the disjointunion ` S ∈ S Hom A ( S, M ) of the sets Hom A ( S, M ), S ∈ S , and choose a well-orderingof this set by identifying it with some ordinal α . So for every ordinal i < α wehave an object S i ∈ S and a morphism f i : S i −→ M . For every ordinal j ≤ α ,put F j = P i Let A be a Grothendieck abelian category. Then there exists aset of acyclic complexes S ⊂ C ( A ) ac such that the class of all acyclic complexes C ( A ) ac ⊂ C ( A ) is the class of all complexes filtered by S , that is C ( A ) ac = Fil ( S ) .Proof. Taking Lemma 8.2 into account, the assertion of the proposition becomes aparticular case of [49, Proposition 4.4]. (cid:3) The following theorem is to be compared with the discussion in [4, Remark 1.3.13]. Theorem 8.4. Let A be a Grothendieck abelian category. Then the pair of classesof objects C ( A ) ac and C ( A inj ) hin is a hereditary complete cotorsion pair in the abeliancategory C ( A ) .Proof. Let S be a set of acyclic complexes in A such that C ( A ) ac = Fil ( S ), as inProposition 8.3. The claim is that the set S ⊂ C ( A ) generates the cotorsion pair weare interested in.Indeed, by Proposition 1.3 we have S ⊥ = ( C ( A ) ac ) ⊥ ⊂ C ( A ). In the pair ofadjoint functors G + : A gr −→ C ( A ) and ( − ) ♯ : C ( A ) −→ A gr , both the functors areexact. Hence of any graded object A ∈ A gr and any complex C • ∈ C ( A ) we haveExt n C ( A ) ( G + ( A ) , C • ) ≃ Ext n A gr ( A, C • ♯ ) for all n ≥ 0. In particular, this isomorphismholds for n = 1. Since the complex G + ( A ) is acyclic, we have shown that ( C ( A ) ac ) ⊥ ⊂ C ( A inj ). Now for a complex J • ∈ C ( A inj ) and any complex X • ∈ C ( A ) we haveExt C ( A ) ( X • , J • ) ≃ Hom K ( A ) ( X • , J • [1]) by Lemma 5.1(a). Thus S ⊥ = ( C ( A ) ac ) ⊥ = C ( A inj ) hin ⊂ C ( A ).Furthermore, the class of all acyclic complexes C ( A ) ac is clearly generating in C ( A ).Applying Theorem 3.4, we can conclude that ⊥ ( C ( A inj ) hin ) = Fil ( S ) ⊕ = C ( A ) ac . ny complex in A is a subcomplex of a contractible complex of injective objects,so the class C ( A inj ) hin is cogenerating in C ( A ). Hence Theorem 3.3 is applicable, andwe can conclude that our cotorsion pair is complete. The cotorsion pair is hereditary,since the class C ( A ) ac is closed under the kernels of epimorphisms in C ( A ). (cid:3) Hovey in [22, Example 3.2] attributes the following corollary to Joyal, who shouldhave mentioned it in a 1984 letter of his to Grothendieck. A weaker version appearedin [2, Theorem 5.4], where existence of enough homotopy injective complexes wasestablished. The existence of enough homotopy injective complexes of injective ob-jects was shown in [46, Theorem 3.13 and Lemma 3.7(ii)] and, following the approachof [22], independently also in [18, Corollary 7.1]. Corollary 8.5. Any Grothendieck abelian category has enough homotopy injectivecomplexes of injective objects. In other words, the natural functor K ( A inj ) hin −→ D ( A ) is a triangulated equivalence.Proof. Here is a proof based on Theorem 8.4. Let A be a Grothendieck abeliancategory. Given a complex C • ∈ C ( A ), we need to find a homotopy injective complexof injective objects J • together with a quasi-isomorphism C • −→ J • of complexesin A . For this purpose, consider a special preenvelope short exact sequence 0 −→ C • −→ J • −→ X • −→ J • ∈ C ( A inj ) hin and X • ∈ C ( A ) ac . Since the complex X • is acyclic, it follows that the monomorphism of complexes C • −→ J • is a quasi-isomorphism. (cid:3) Theorem 8.6. Let A be a Grothendieck abelian category. Then the triple of classesof objects L = C ( A ) , W = C ( A ) ac , and R = C ( A inj ) hin is a cofibrantly generatedhereditary abelian model structure on the abelian category of complexes C ( A ) .Proof. The pair of classes ( W , R ) is a complete cotorsion pair in C ( A ) by Theorem 8.4.The abelian category of complexes C ( A ) has enough injective objects since it isGrothendieck (or since the abelian category A has). According to Lemma 5.3(a),it follows that R ∩ W = C ( A ) inj . The class of acyclic complexes W is obviously thickin C ( A ) (see also Lemma 4.6(a) or 5.3(a)). By Lemma 4.5(a) (applied to the cate-gory C ( A )), the triple ( L , W , R ) is an injective abelian model structure on C ( A ).As explained in Section 4, any injective abelian model structure is hereditary.Finally, it is clear from the proof of Theorem 8.4 that the cotorsion pair ( W , R ) is C ( A ) is generated by a set of objects. The cotorsion pair ( C ( A ) , C ( A ) inj ) is generatedby any set of complexes S such that C ( A ) = Fil ( S ), as in Lemma 8.2 applied tothe category C ( A ). According to Corollary 4.3, this means that our abelian modelstructure is cofibrantly generated. (cid:3) The abelian model structure ( L , W , R ) defined in Theorem 8.6 is called the injectivederived model structure on the abelian category of complexes C ( A ). Lemma 8.7. For any Grothendieck abelian category A , the class W of all weakequivalences in the injective derived model structure on the abelian category C ( A ) coincides with the class of all quasi-isomorphisms of complexes in A . roof. Similar to Lemma 6.9. (cid:3) The following corollary presumes existence of (set-indexed) coproducts in the de-rived category D ( A ). Notice that, since the coproducts are exact in the abelian cate-gory A and consequently the thick subcategory of acyclic complexes K ( A ) ac ⊂ K ( A ) isclosed under coproducts, the coproducts in the derived category D ( A ) are induced bythose in the homotopy category K ( A ). In other words, the Verdier quotient functor K ( A ) −→ D ( A ) preserves coproducts [31, Lemma 3.2.10]. Corollary 8.8. For any Grothendieck abelian category A , the (unbounded) derivedcategory D ( A ) is a well-generated triangulated category.Proof. This result can be found in [27, Section 7.7], or in a more precise form in [28,Theorem 5.10]. It is also provable similarly to Corollary 6.10. (cid:3) The next theorem appeared in the context of DG-comodules over a DG-coalgebra(over a field) in the postpublication arXiv version of [35, Section 5.5]; see also [36,Theorem 1.1(d)]. Theorem 8.9. Assume Vopˇenka’s principle, and let A be a Grothendieck abeliancategory with an injective cogenerator J . Then the category D ( A ) is generated, asa triangulated category with products, by the single object J (viewed as a one-termcomplex concentrated in degree ). In other words, the full subcategory of homotopyinjective complexes K ( A ) hin ⊂ K ( A ) is the minimal strictly full triangulated subcate-gory in K ( A ) containing the object J and closed under products.Proof. Notice first of all that (set-indexed) products in the derived category D ( A )can be simply constructed as the products in the homotopy category K ( A ) hin , whichis equivalent to D ( A ) by Corollary 8.5. The full subcategory K ( A ) hin is closed underproducts in K ( A ), so the products in K ( A ) hin (just as in K ( A )) can be computed asthe termwise products of complexes.The key observation is that Hom D ( A ) ( X • , J [ i ]) = 0 for all i ∈ Z implies X • = 0 fora given object X • ∈ D ( A ), since for any complex X • one has a natural isomorphismof abelian groups Hom D ( A ) ( X • , J [ i ]) ≃ Hom A ( H − i ( X • ) , J ). Denote by D ′′ ⊂ D ( A ) theminimal triangulated subcategory in D ( A ) containing J and closed under products.Following [4, Corollary 1.1.15 and the preceding discussion], any hereditary abelianmodel category is stable. The derived category D ( A ) can be equivalently defined asthe homotopy category C ( A )[ W − ] of the injective derived model category structureon C ( A ). The category C ( A ) is Grothendieck, hence locally presentable. Accordingto [10, Theorem 2.4], assuming Vopˇenka’s principle, any triangulated subcategoryclosed under products in D ( A ) is reflective. So, in particular, the inclusion functor D ′′ −→ D ( A ) has a left adjoint.The left adjoint functor to a fully faithful triangulated functor is a Verdier quotientfunctor. The kernel of our functor D ( A ) −→ D ′′ consists of complexes X • satisfyingHom D ( A ) ( X • , J [ i ]) = 0 for all i ∈ Z , since J [ i ] ∈ D ′′ . Thus all such objects X • ∈ D ( A )vanish, and the functor D ( A ) −→ D ′′ is a triangulated equivalence. It follows that he inclusion D ′′ −→ D ( A ) is a triangulated equivalence, too; so D ′′ = D ( A ), asdesired. (cid:3) Coderived Model Structure In this section we consider coderived categories in the sense of Becker [4]. Inwell-behaved cases, this means simply the homotopy category of complexes of injec-tive objects (which was studied first by Krause [26] in the case of locally NoetherianGrothendieck categories); see the discussion in the introduction. The coderived cat-egory in the sense of Becker was also considered in the preprint [51, Section 6].The coderived category in the sense of Becker has to be distinguished from thecoderived category in the sense of the books and papers [34, 35, 37, 14, 43, 38]. Thetwo definitions of a coderived category are known to be equivalent under certainassumptions [35, Theorem 3.7], but it is still an open question whether they areequivalent for the category of modules over an arbitrary associative ring (see [38,Example 2.5(3)] for a discussion).Let A be an abelian category. A complex X • ∈ K ( A ) is said to be coacyclic (in the sense of Becker) if Hom K ( A ) ( X • , J • ) = 0 for any complex of injective objects J • ∈ K ( A inj ). We denote the full subcategory of coacyclic complexes by K ( A ) coac ⊂ K ( A )and its full preimage under the natural functor C ( A ) −→ K ( A ) by C coac ( A ). Clearly, K ( A ) coac is a triangulated (and even thick) subcategory in the homotopy category K ( A ).The quotient category D co ( A ) = K ( A ) / K ( A ) coac is called the coderived category of A (in the sense of Becker). Lemma 9.1. (a) For any short exact sequence −→ K • −→ L • −→ M • −→ ofcomplexes in A , the total complex Tot( K • → L • → M • ) of the bicomplex with threerows K • −→ L • −→ M • belongs to K ( A ) coac . (b) The full subcategory of coacyclic complexes K ( A ) coac is closed under coproductsin the homotopy category K ( A ) .Proof. This is our version of [35, Theorem 3.5(a)]. It can be obtained from Lemma 7.1by inverting the arrows. (cid:3) Theorem 9.2. Let A be a Grothendieck abelian category. Then the pair of classesof objects C ( A ) coac and C ( A inj ) is a hereditary complete cotorsion pair in the abeliancategory C ( A ) .Proof. Let S be a set of objects in A such that A = Fil ( S ), as in Lemma 8.2. Theclaim is that the cotorsion pair we are interested in is generated by the set of two-term complexes S = { G + ( S )[ i ] } S ∈ S , i ∈ Z ⊂ C ( A ) (where S ∈ S is viewed as a gradedobject concentrated in degree 0).Indeed, for any graded object A ∈ A gr and any complex C • ∈ C ( A ) we haveExt C ( A ) ( G + ( A ) , C • ) ≃ Ext A gr ( A, C • ♯ ), as explained in the proof of Theorem 8.4.In particular, for S ∈ S we have Ext C ( A ) ( G + ( S )[ i ] , C • ) ≃ Ext A gr ( S [ i ] , C • ♯ ) ≃ Ext A ( S, C − i ). In view of Proposition 1.3, it follows that S ⊥ = C ( A inj ) ⊂ C ( A ). urthermore, for any complex X • ∈ C ( A ) and any complex of injective objects J • ∈ C ( A ) we have Ext C ( A ) ( X • , J • ) ≃ Hom K ( A ) ( X • , J • [1]) by Lemma 5.1(a). Therefore, ⊥ ( C ( A inj )) = C ( A ) coac . Hence our pair of classes of objects is indeed the cotorsion pairgenerated by the set S ⊂ C ( A ).Any complex is a quotient complex of a contractible complex. All contractiblecomplexes are coacyclic; so the class C ( A ) coac is generating in C ( A ). Any complexin A is also a subcomplex of a complex of injective objects, so the class C ( A inj ) iscogenerating in C ( A ). Applying Theorem 3.3, we conclude that our cotorsion pair iscomplete. The cotorsion pair is hereditary, since the class C ( A ) coac is closed under thekernels of epimorphisms in C ( A ), as one can see from Lemma 9.1(a). It is also clearthat the class C ( A inj ) ⊂ C ( A ) is closed under the cokernels of monomorphisms. (cid:3) The proof of Theorem 9.2 also provides us with the following description of theclass of coacyclic complexes in the sense of Becker. Corollary 9.3. Let A be a Grothendieck category. Then the coacyclic complexesare precisely direct summands of those filtered by contractible complexes. In fact, if S ⊂ A is a set of objects such that A = Fil ( S ) and S = { G + ( S )[ i ] } S ∈ S , i ∈ Z ⊂ C ( A ) as before, then C ( A ) coac = Fil ( S ) ⊕ .Proof. Let L ∈ A be a generator. Since L ∈ Fil ( S ), we clearly have G + ( L )[ i ] ∈ Fil ( S )for each i ∈ Z . Since Fil ( S ) is also closed under coproducts in C ( A ), it is a generatingclass and, thus, Theorem 3.4 tells that C ( A ) coac = Fil ( S ) ⊕ . Since every complex in S iscontractible and any contractible complex is coacyclic, it also follows that C ( A ) coac isthe class of all direct summands of complexes filtered by contractible complexes. (cid:3) Corollary 9.4. For any Grothendieck abelian category A , the composition of theinclusion of triangulated categories K ( A inj ) −→ K ( A ) with the Verdier quotient functor K ( A ) −→ D co ( A ) is a triangulated equivalence K ( A inj ) ≃ D co ( A ) .Proof. It is clear from the definitions that the functor K ( A inj ) −→ D co ( A ) is fullyfaithful (use the argument dual to the proof of Proposition 6.4). In order to provethe corollary, it remains to find, for any complex C • ∈ K ( A ), a complex of injectiveobjects J • ∈ K ( A inj ) together with a morphism of complexes C • −→ J • whose conebelongs to C ( A ) coac .For this purpose, consider a special preenvelope short exact sequence 0 −→ C • −→ J • −→ X • −→ J • ∈ C ( A inj ) and X • ∈ C ( A ) coac . By Lemma 9.1, the totalization Tot( C • −→ J • −→ X • )of the short exact sequence 0 −→ C • −→ J • −→ X • −→ X • is coacyclic, it follows that the cone of the morphism C • −→ J • is coacyclic, too. (cid:3) In terms of the full subcategory K ( A inj ) ⊂ K ( A ), Corollary 9.4 means that thetriangulated inclusion functor K ( A inj ) −→ K ( A ) has a left adjoint. This result can bealso directly deduced from [8, Theorem 3.5] and can be found in [33, Theorem 2.13]or [28, Corollary 5.13]. heorem 9.5. Let A be a Grothendieck abelian category. Then the triple of classes ofobjects L = C ( A ) , W = C ( A ) coac , and R = C ( A inj ) is a cofibrantly generated hereditaryabelian model structure on the abelian category of complexes C ( A ) .Proof. This is similar to the proof of Theorem 8.6. The pair of classes ( W , R ) is acomplete cotorsion pair in C ( A ) by Theorem 9.2. According to Lemma 5.3(a), itfollows that R ∩ W = C ( A ) inj . It follows from Lemma 9.1(a) that the class of coacycliccomplexes W is thick in C ( A ) (see also Lemma 4.6(a) or 5.3(a)). By Lemma 4.5(a),the triple ( L , W , R ) is an injective abelian model structure on the category C ( A ).Finally, it was shown in the proof of Theorem 9.2 that the cotorsion pair ( L , W ) in C ( A ) is generated by a set of objects. For the cotorsion pair ( C ( A ) , C ( A ) inj ), the samewas explained in the proof of Theorem 8.6. (cid:3) The abelian model structure ( L , W , R ) defined in Theorem 9.5 is called the coderivedmodel structure on the abelian category of complexes C ( A ). Lemma 9.6. For any Grothendieck abelian category A , the class W of all weakequivalences in the coderived model structure on the abelian category C ( A ) coincideswith the class of all morphisms of complexes in A with the cones belonging to C ( A ) coac .Proof. Similar to the proof of Lemma 7.6. One needs to notice that, by Lemma 9.1(a),a monomorphism of complexes in C ( A ) has coacyclic cokernel if and only if it hascoacyclic cone, and similarly, an epimorphism of complexes in C ( A ) has coacyclickernel if and only if it has coacyclic cone. (cid:3) The following corollary presumes existence of (set-indexed) coproducts in thecoderived category D co ( A ). Here we notice that, since the thick subcategory ofcoacyclic complexes K ( A ) coac ⊂ K ( A ) is closed under coproducts by Lemma 9.1(b), thecoproducts in the coderived category D co ( A ) are induced by those in the homotopycategory K ( A ). In other words, the Verdier quotient functor K ( A ) −→ D co ( A )preserves coproducts [31, Lemma 3.2.10]. Corollary 9.7. For any Grothendieck abelian category A , the coderived category D co ( A ) is a well-generated triangulated category.Proof. This result can be found in [33, Theorem 3.13] or [28, Theorem 5.12]. It is alsoprovable similarly to Corollary 7.7. One notices that the coderived category D co ( A )can be equivalently defined as the homotopy category C ( A )[ W − ] of the coderivedmodel structure on C ( A ) and uses the fact that the homotopy category of any stablecombinatorial model category is well-generated. (cid:3) Remark 9.8. One can consider the possibility of extending the results of Sections 8–9to locally presentable abelian categories A with enough injective objects . A specificproblem arising in this connection is that it is not clear how to prove a generalizationof Lemma 8.2 not depending on the assumption that the directed colimits are exactin A . Why is the directed colimit of the chain of subobjects ( F i ) i In this section, we consider exact categories in the sense of Quillen. An exact cate-gory is an additive category endowed with a class of short exact sequences satisfyingnatural axioms. For a reference, see [9].A morphism A −→ B in an exact category E is said to be an admissible monomor-phism if there exists a short exact sequence 0 −→ A −→ B −→ C −→ 0. Themorphism B −→ C is said to be an admissible epimorphism in this case. We willalso say that A is an admissible subobject in B .An additive category E is said to be weakly idempotent-complete if for every pairof morphisms i : A −→ B and p : B −→ A in E such that pi = id A there exists anobject C ∈ E and an isomorphism B ≃ A ⊕ C transforming the morphism i intothe direct summand inclusion A −→ A ⊕ C and the morphism p into the directsummand projection A ⊕ C −→ A . We will assume our exact category E to beweakly idempotent-complete (this assumption simplifies the theory of exact categoriesconsiderably [9, Section 7]).Let E be an exact category. We will consider a function ψ assigning to every object E ∈ E a regular cardinal ψ ( E ). Given a regular cardinal λ , denote by E λ ⊂ E the classof all objects E ∈ E such that ψ ( E ) ≤ λ . Instead of the function ψ , one can considerthe induced filtration of the category E by the full subcategories E λ . Obviously, onehas E λ ⊂ E µ for all λ ≤ µ and E = S λ E λ .The following conditions are imposed on the function ψ , or equivalently, on thefull subcategories E λ ⊂ E :(i) ψ (0) = 0, or in other words, 0 ∈ E ; for any two isomorphic objects E ′ and E ′′ in E , one has ψ ( E ′ ) = ψ ( E ′′ );(ii) for every regular cardinal λ , there is only a set of objects, up to isomorphism,in the full subcategory E λ ⊂ E ;(iii) for every regular cardinal λ , the full subcategory E λ is closed under extensionsin E , that is, in other words, if 0 −→ A −→ B −→ C −→ E and A , C ∈ E λ , then B ∈ E λ ;(iv) for any regular cardinal λ , an object B ∈ E , and a collection of less than λ morphisms ( A i → B ) i ∈ I with the domains A i ∈ E λ , there exists an admissiblesubobject D ⊂ B with D ∈ E λ such that the morphism A i −→ B factorizesthrough the admissible monomorphism D −→ B for every i ∈ I ;(v) for any regular cardinal λ and any admissible epimorphism B −→ C with C ∈ E λ there exists an admissible subobject D ⊂ B such that D ∈ E λ andthe composition D −→ B −→ C is an admissible epimorphism. e will say that ψ is an object size function on E if the conditions (i–v) are satisfied.It follows from the conditions (i) and (iii) that the full subcategory E λ ⊂ E inheritsan exact category structure from the exact category E . Specifically, a composablepair of morphisms in E λ is said to be a short exact sequence in E λ if it is a short exactsequence in E . Proposition 10.1. Any locally presentable abelian category E with its abelian exactstructure admits an object size function.Proof. Let E be a locally κ -presentable abelian category (where κ is a regular cardi-nal). Define a function ψ on the objects E ∈ E by the rules ψ (0) = 0, ψ ( E ) = κ if E = 0 is κ -generated, and ψ ( E ) = λ if λ ≥ κ is the minimal regular cardinal forwhich E is λ -generated.Here an object E ∈ E is said to be λ -generated for a regular cardinal λ if thefunctor Hom E ( E, − ) : E −→ Sets preserves the colimits of λ -directed diagrams ofmonomorphisms [1, Section 1.E]. An object E ∈ E is λ -generated for a given λ ≥ κ if and only if E is a quotient of a λ -presentable object [1, Proposition 1.69].Then condition (i) is satisfied by construction, and condition (ii) holds by [1, Corol-lary 1.69]. To check the remaining conditions, the following lemma will be useful. Lemma 10.2. Let λ be a regular cardinal and E be a locally λ -presentable abeliancategory. Then an object E ∈ E is λ -generated if and only if it cannot be presentedas the sum of a λ -directed set of its proper subobjects (in the inclusion order).Proof. The key observation is that, in a locally λ -presentable category, λ -directedcolimits commute with λ -small (in particular, finite) limits [1, Proposition 1.59]. Foran abelian locally λ -presentable category E , this means that the functors of λ -directedcolimit are exact. We will use the facts that λ -directed colimits in E commute withthe kernels and pullbacks.“Only if”: Let ( E i ⊂ E ) i ∈ I be a λ -directed family of subobjects in E , orderedby inclusion. Then the natural morphism lim −→ i ∈ I E i −→ E is injective, since themorphisms E i −→ E are injective and the functor lim −→ i ∈ I commutes with the kernels.It follows that lim −→ i ∈ I E i = P i ∈ I E i ⊂ E .Now assume that E = lim −→ i ∈ I E i . Then, since the object E is λ -generated, theidentity morphism E −→ lim −→ i ∈ I E i = E factorizes through the object E i for someindex i ∈ I . Thus E i = E .“If”: Let ( f ij : F i −→ F j ) i Let κ < λ be two infinite regular cardinals. Let E be a weaklyidempotent-complete exact category with exact colimits of κ -directed chains of admis-sible monomorphisms and an object size function ψ preserved by such colimits. Put µ = φ κ, E ( λ ) . Let K • be a complex in E λ , let M • be an exact complex in E , and let g : K • −→ M • be a morphism of complexes in E . Then there exist an exact complex L • in E µ and morphisms of complexes k : K • −→ L • and m : L • −→ M • such that g = mk and the component m n : L n −→ M n of the morphism of complexes m in thedegree n is an admissible monomorphism in E for every n ∈ Z .Proof. Let N n ∈ E be objects for which there exist short exact sequences 0 −→ N n −→ M n −→ N n +1 −→ E such that the differential M n −→ M n +1 is equalto the composition M n −→ N n +1 −→ M n +1 for every n ∈ Z . Consider the diagramformed by the morphisms N n −→ M n and M n −→ N n +1 in the category E .Proceeding by transfinite induction, we will construct a chain of admissible sub-diagrams ( N ni → M ni → N n +1 i ) n ∈ Z in the diagram ( N n → M n → N n +1 ) n ∈ Z indexedby the ordinals 0 ≤ i ≤ κ . This means that for every 0 ≤ i ≤ κ we will produceadmissible subobjects M ni ⊂ M n and N ni ⊂ N n such that M ni ⊂ M nj and N ni ⊂ N nj for all 0 ≤ i ≤ j ≤ κ , and the morphisms N n −→ M n and M n −→ N n +1 take N ni into M ni and M ni into N n +1 i . Furthermore, the morphisms g n : K n −→ M n ill factorize through the admissible monomorphisms M n −→ M n . Finally, wewill have M ni , N ni ∈ E φ i ( λ ) for all 0 ≤ i ≤ κ , n ∈ Z , and the short sequences0 −→ N nκ −→ M nκ −→ N n +1 κ −→ E (equivalently, in E φ i ( λ ) ) for all n ∈ Z . Then we will put L • = M • κ , that is L n = M nκ for every n ∈ Z .Put M n = 0 = N n for all n ∈ Z . To construct the admissible subobjects M n ⊂ M n and N n ⊂ N n for every n ∈ Z , we start with the compositions K n −→ M n −→ N n +1 of the morphisms g n : K n −→ M n with the admissible epimorphisms M n −→ N n +1 . Using property (iv), we choose for every n an admissible subobject D n +11 ⊂ N n +1 such that D n +11 ∈ E λ and the morphism K n −→ N n +1 factorizes through theadmissible monomorphism D n +11 −→ N n +1 . Let F n denote the pullback as depictedin the following commutative diagram with short exact sequences in the rows and(necessarily) admissible monomorphisms in the columns:0 / / N n / / F n / / (cid:15) (cid:15) D n +11 / / (cid:15) (cid:15) / / N n / / M n / / N n +1 / / G n ⊂ F n such that G n ∈ E λ and the composition G n −→ F n −→ D n +11 is an admissible epimorphism.Notice that the morphism g n : K n −→ M n factorizes through the admissiblemonomorphism F n −→ M n , since the composition K n −→ M n −→ N n +1 factor-izes through the admissible monomorphism D n +11 −→ N n +1 . Using property (iv),we choose an admissible subobject M n ⊂ F n such that M n ∈ E λ and the threemorphisms G n −→ F n , K n −→ F n , and D n −→ N n −→ F n factorize through theadmissible monomorphism M n −→ F n . The composition of admissible monomor-phisms M n −→ F n −→ M n is an admissible monomorphism. The composition G n −→ M n −→ F n −→ D n +11 is an admissible epimorphism by construction;since the exact category E is weakly idempotent-complete, it follows that the mor-phism M n −→ D n +11 is an admissible epimorphism [9, Proposition 7.6]. Let N n denote the kernel of the latter admissible epimorphism; then N n ∈ E φ ( λ ) . More-over, since the composition N n −→ M n −→ M n is an admissible monomorphismby the construction and E is weakly idempotent-complete, an application of [9,Proposition 7.6] to E op shows that N n −→ N n is an admissible monomorphism.Since the composition D n −→ N n −→ M n −→ N n +1 vanishes, and hence so does D n −→ M n −→ F n −→ D n +11 , the map D n −→ M n factors through N n −→ M n and the factorization D n −→ N n is an admissible monomorphism using the sameargument as above.To summarize, we have the following commutative diagram with short exact se-quences in the rows and all the vertical maps being admissible monomorphisms for ll n ∈ Z , D n (cid:15) (cid:15) K n ∃ ! | | g n ☛☛☛☛☛☛☛☛ (cid:5) (cid:5) ☛☛☛☛☛☛☛☛ / / N n / / (cid:15) (cid:15) M n / / (cid:15) (cid:15) D n +11 / / (cid:15) (cid:15) / / N n / / M n / / N n +1 / / , and we have constructed the desired admissible subdiagram ( N n → M n → N n +11 ) n ∈ Z in the diagram ( N n → M n → N n +1 ) n ∈ Z . The first step of our transfinite inductionis complete.Let 2 ≤ j < κ be an ordinal. Suppose that we have already constructed theadmissible subdiagrams ( N ni → M ni → N n +1 i ) n ∈ Z in the diagram ( N n → M n → N n +1 ) n ∈ Z for all the ordinals i < j . Using property (iv), choose for every n ∈ Z anadmissible subobject D nj ⊂ N n containing all the admissible subobjects N ni ⊂ N n for i < j and such that D nj ∈ E φ j ( λ ) . We again construct the following pullbackdiagram with short exact sequences in the rows and admissible monomorphisms inthe columns, 0 / / N n / / F nj / / (cid:15) (cid:15) D n +1 j / / (cid:15) (cid:15) / / N n / / M n / / N n +1 / / G nj ⊂ F nj such that G nj ∈ E φ j ( λ ) and the composition G nj −→ F nj −→ D n +1 j is an admissible epimorphism. Usingproperty (iv), choose an admissible subobject M nj ⊂ F nj such that M nj ∈ E φ j ( λ ) and the two morphisms G nj −→ F nj and D nj −→ N n −→ F nj factorize through theadmissible monomorphism M nj −→ F nj . The composition of admissible monomor-phisms M n −→ F n −→ M n is an admissible monomorphism. The composition G nj −→ M nj −→ F nj −→ D n +1 j is an admissible epimorphism by construction; by [9,Proposition 7.6], it follows that the morphism M nj −→ D n +1 j is an admissible epimor-phism. Let N nj denote its kernel; then N nj ∈ E φ j ( λ ) .Arguing as in the case of j = 1 above, one shows that N nj −→ N n is an admissiblemonomorphism and D nj ⊂ N nj ⊂ N n . This finishes the construction of the admissiblesubdiagrams ( N nj → M nj → N n +1 j ) n ∈ Z in the diagram ( N n → M n → N n +1 ) n ∈ Z for allordinals 0 ≤ j < κ . In particular, we have the following commutative diagram withshort exact sequences in the rows and admissible monomorphisms in the columns for ach n ∈ Z and i < j D nj (cid:15) (cid:15) N n +1 i (cid:15) (cid:15) / / N nj / / (cid:15) (cid:15) M nj / / (cid:15) (cid:15) D n +1 j / / (cid:15) (cid:15) / / N n / / M n / / N n +1 / / , Using the assertion dual to [9, Proposition 7.6] again, the inclusions of admissiblesubobjects M ni −→ M nj and N ni −→ N nj are admissible monomorphisms for all 0 ≤ i < j < κ . Thus ( M ni ) ≤ i<κ and ( N ni ) ≤ i<κ are chains of admissible monomorphisms.By condition (vi), it follows that the colimits M nκ = lim −→ i<κ M ni and N nκ = lim −→ i<κ N ni exist. By condition (vii), the natural morphisms M nκ −→ M n and N nκ −→ N n areadmissible monomorphisms. By condition (viii), we have M nκ , N nκ ∈ E φ κ ( λ ) = E µ . Wehave constructed the admissible subdiagram ( N nκ → M nκ → N n +1 κ ) n ∈ Z in the diagram( N n → M n → N n +1 ) n ∈ Z .In order to show that the complex L • = M • κ is exact, it remains to check exactnessof the short sequences 0 −→ N nκ −→ M nκ −→ N n +1 κ −→ 0. Here we observe that( D ni ) ≤ i<κ is a chain of admissible subobjects in N n mutually cofinal with the chain( N ni ) ≤ i<κ . Hence lim −→ i<κ D ni = N nκ . Applying condition (vii) to the chain of admis-sible monomorphisms of short exact sequences 0 −→ N ni −→ M ni −→ D n +1 i −→ ≤ i ≤ κ , we conclude that 0 −→ lim −→ i<κ N ni −→ lim −→ i<κ M ni −→ lim −→ i<κ D n +1 i −→ E . (cid:3) We refer to [9, Section 10.4] for the definition of the derived category D ( E ) of anexact category E as the Verdier quotient category of the homotopy category K ( E ) bythe triangulated subcategory of exact complexes. We also refer to [54, Set-TheoreticRemark 10.3.3] for a discussion of “existence” of localizations of categories (includingtriangulated Verdier quotient categories, such as D ( E )). In a different terminology,the question is whether the derived category D ( E ) “has Hom sets”. Corollary 10.4. Let E be a weakly idempotent-complete exact category with exactcolimits of κ -directed chains of admissible monomorphisms and an object size functionpreserved by such colimits, for some regular cardinal κ . Then the derived category D ( E ) “exists” or “has Hom sets”, in the sense that morphisms between any given twoobjects in D ( E ) form a set rather than a proper class.Proof. It is clear from Theorem 10.3 that for every complex K • ∈ K ( E ) there exists aset of morphisms f t : K • −→ L • t , t ∈ T from the complex K • to exact complexes L • t in K ( E ) such that every morphism from K • to an exact complex M • in K ( E ) factorizesthrough one of the morphisms f t . It follows that the multiplicative system S of allmorphisms with exact cones in K ( E ) is locally small in the sense of [54, Set-TheoreticConsiderations 10.3.6]; hence the localization D ( E ) = K ( E )[ S − ] exists. (cid:3) heorem 10.5. Let E be a locally presentable abelian category. Then the derivedcategory D ( E ) “exists”, in the sense that morphisms between any given two objectsin D ( E ) form a set rather than a proper class.Proof. Let κ be a regular cardinal for which the category E is locally κ -presentable.Then condition (vi) is satisfied, since all colimits exist in E . Furthermore, condi-tion (vii) holds, because the functors of κ -directed colimit are exact in E [1, Proposi-tion 1.59]. A construction of an object size function ψ on E is spelled out in Proposi-tion 10.1. So conditions (i–v) are satisfied as well. Finally, condition (viii) is provablesimilarly to the proof of condition (iv) in Proposition 10.1. Thus Corollary 10.4 isapplicable, and we are done. (cid:3) Question 10.6. Let E be a locally presentable abelian category.(a) Do set-indexed coproducts necessarily exist in the derived category D ( E ) ?(b) If the answer to (a) is positive, is then the triangulated category D ( E ) well-generated? References [1] J. Ad´amek, J. Rosick´y. Locally presentable and accessible categories. London Math. SocietyLecture Note Series 189, Cambridge University Press, 1994.[2] L. Alonso Tarr´ıo, A. Jerem´ıas L´opez, M. J. Souto Salorio. Localization in categories of com-plexes and unbounded resolutions. Canadian Journ. of Math. , Journ. of Pureand Appl. Algebra , arXiv:1907.04973 [math.RA] [4] H. Becker. Models for singularity categories. Advances in Math. , p. 187–232, 2014. arXiv:1205.4473 [math.CT] [5] T. Beke. Sheafifiable homotopy model categories. Math. Proc. Cambridge Phil. Soc. , Mem. Amer.Math. Soc. , Lecture Notes in Math. ,Springer-Verlag, Berlin, 1994.[8] D. Bravo, E. E. Enochs, A. C. Iacob, O. M. G. Jenda, J. Rada, Cortorsion pairs in C ( R - Mod ). Rocky Mountain J. Math. , Expositiones Math. , arXiv:0811.1480[math.HO] [10] C. Casacuberta, J. J. Guti´errez, J. Rosick´y. Are all localizing subcategories of stable homo-topy categories coreflective? Advances in Math. , p. 158–184, 2014. arXiv:1106.2218[math.CT] [11] C. Casacuberta, J. Rosick´y. Combinatorial homotopy categories. Bousfield Classes andOhkawa’s Theorem , Nagoya, August 2015, Springer Proceedings in Mathematics & Statistics309, Springer, 2020, pp. 89–101. arXiv:1702.00240 [math.AT] [12] R. Colpi, K. R. Fuller. Tilting objects in abelian categories and quasitilted rings. Trans. Amer.Math. Soc. , Math. Z. , arXiv:1707.01677 [math.AG] 14] A. I. Efimov, L. Positselski. Coherent analogues of matrix factorizations and relative singu-larity categories. Algebra and Number Theory , arXiv:1102.0261[math.CT] [15] P. C. Eklof, J. Trlifaj. How to make Ext vanish. Bulletin of the London Math. Society , Math. Zeitschrift , Journ. of Pure and Appl. Algebra , arXiv:1009.3574 [math.AT] [20] A. Grothendieck. Sur quelques points d’alg`ebre homologique. Tohoku Math. Journ. (2) , Math.Zeitschrift , Advances in Math. , arXiv:math.RA/0312088 [24] B. Keller. Deriving DG-categories. Ann. Sci. de l’ ´Ecole Norm. Sup. (4) , Documenta Math. , p. 121–126, 2001.[26] H. Krause. The stable derived category of a Noetherian scheme. Compositio Math. , arXiv:math.AG/0403526 [27] H. Krause. Localization theory for triangulated categories. In: Th. Holm, P. Jørgensen andR. Rouquier, Eds., Triangulated categories , London Math. Lecture Note Ser. 375, CambridgeUniversity Press, 2010, p. 161–235. arXiv:0806.1324 [math.CT] [28] H. Krause. Deriving Auslander’s formula. Documenta Math. , p. 669–688, 2015. arXiv:1409.7051 [math.CT] [29] J. Lurie. Higher topos theory. Annals of Math. Studies , 170, Princeton Univ. Press, 2009.xviii+925 pp.[30] M. Makkai, J. Rosick´y. Cellular categories. Journ. of Pure and Appl. Algebra , arXiv:1304.7572 [math.CT] [31] A. Neeman. Triangulated categories. Annals of Math. Studies, Princeton Univ. Press, 2001.viii+449 pp.[32] A. Neeman. The homotopy category of flat modules, and Grothendieck duality. InventionesMath. , Algebra and Number Theory , arXiv:0708.3398 [math.CT] [35] L. Positselski. Two kinds of derived categories, Koszul duality, and comodule-contramodulecorrespondence. Memoirs of the American Math. Society , arXiv:0905.2621 [math.CT] 36] L. Positselski. Dedualizing complexes of bicomodules and MGM duality over coalgebras. Alge-bras and Represent. Theory , arXiv:1607.03066 [math.CT] [37] L. Positselski. Weakly curved A ∞ -algebras over a topological local ring. M´emoires de la Soci´et´eMath´ematique de France , 2018. vi+206 pp. arXiv:1202.2697 [math.CT] [38] L. Positselski. Pseudo-dualizing complexes and pseudo-derived categories. Rendiconti SeminarioMatematico Univ. Padova , p. 153–225, 2020. arXiv:1703.04266 [math.CT] [39] L. Positselski. Abelian right perpendicular subcategories in module categories. Electronicpreprint arXiv:1705.04960 [math.CT] .[40] L. Positselski. Remarks on derived complete modules and complexes. Electronic preprint arXiv:2002.12331 [math.AC] .[41] L. Positselski, J. Rosick´y. Covers, envelopes, and cotorsion theories in locally presentableabelian categories and contramodule categories. Journ. of Algebra , p. 83–128, 2017. arXiv:1512.08119 [math.CT] [42] L. Positselski, J. ˇSt’ov´ıˇcek. The tilting-cotilting correspondence. Internat. Math. Research No-tices , arXiv:1710.02230 [math.CT] [43] L. Positselski, J. ˇSt’ov´ıˇcek. ∞ -tilting theory. Pacific Journ. of Math. , arXiv:1711.06169 [math.CT] [44] J. Rosick´y. Generalized Brown representability in homotopy categories. Theory and Appl. ofCategories , arXiv:math/0506168 [math.CT] [45] L. Salce. Cotorsion theories for abelian groups. Symposia Math. XXIII , Academic Press,London–New York, 1979, p. 11–32.[46] C. Serp´e. Resolution of unbounded complexes in Grothendieck categories. Journ. of Pure andAppl. Algebra , Advances in Math. , arXiv:1005.3248 [math.CT] [48] N. Spaltenstein. Resolutions of unbounded complexes. Compositio Math. , Forum Math. , arXiv:1005.3251 [math.CT] [50] J. ˇSt’ov´ıˇcek. Exact model categories, approximation theory, and cohomology of quasi-coherentsheaves. Advances in representation theory of algebras , p. 297–367, EMS Ser. Congr. Rep., Eur.Math. Soc., Z¨urich, 2013. arXiv:1301.5206 [math.CT] [51] J. ˇSt’ov´ıˇcek. On purity and applications to coderived and singularity categories. Electronicpreprint arXiv:1412.1615 [math.CT] .[52] J.-L. Verdier. Cat´egories d´eriv´ees, ´etat 0. SGA 4 1/2. Lecture Notes in Math. , p. 262–311,1977.[53] J.-L. Verdier. Des cat´egories d´eriv´ees des cat´egories ab´eliennes. Ast´erisque , 1996.[54] C. A. Weibel. An introduction to homological algebra. Cambridge Studies in Advanced Math-ematics, 38. Cambridge University Press, 1994.(Leonid Positselski) Institute of Mathematics of the Czech Academy of Sciences,ˇZitn´a 25, 115 67 Prague 1, Czech Republic; andLaboratory of Algebra and Number Theory, Institute for Information Transmis-sion Problems, Moscow 127051, Russia Email address : [email protected] (Jan ˇSt’ov´ıˇcek) Charles University in Prague, Faculty of Mathematics and Physics,Department of Algebra, Sokolovsk´a 83, 186 75 Praha, Czech Republic Email address : [email protected]@karlin.mff.cuni.cz