Derived equivalences of functor categories

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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa

Derived equivalences of functor categories ✩ Javad Asadollahi a,b,∗ , Rasool Hafezi b , Razieh Vahed c,b a

Department of Mathematics, University of Isfahan, P.O. Box: 81746-73441, Isfahan, Iran School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O. Box: 19395-5746, Tehran, Iran c Department of Mathematics, University of Khansar, P.O. Box: 87916-85163, Khansar, Iran b

a r t i c l e

i n f o

Article history: Received 14 August 2016 Received in revised form 22 March 2018 Available online xxxx Communicated by S. Koenig MSC: 18E30; 16E35; 16E65; 16G10

a b s t r a c t Let Mod-S denote the category of S-modules, where S is a small pre-additive category. Using the notion of relative derived categories of functor categories, we generalize Rickard’s theorem on derived equivalences of module categories over rings to Mod-S. Several interesting applications will be provided. In particular, it will be shown that derived equivalence of two coherent rings not only implies the equivalence of their homotopy categories of projective modules, but also implies that they are Gorenstein derived equivalent. As another application, it is shown that a good tilting module produces an equivalence between the unbounded derived category of the module category of the ring and the relative derived category of the module category of the endomorphism ring of it. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Throughout the paper all categories are assumed to be pre-additive. Let S be a small category. We denote by Mod-S the category of all contravariant functors from S to Ab, the category of abelian groups. Mod-S, known also as functor category, will be called the category of modules on S. It is known that it is an abelian category having enough projective objects. Let K(Mod-S) denote the homotopy category of Mod-S and fix a set of objects T of perfect complexes over Mod-S satisfying properties (P1) and (P2) of 3.1 below. A complex X  in K(Mod-S) is said to be T -acyclic, if for every object T  ∈ T the induced complex HomT (T  , X  ) is acyclic. Let KT -ac (Mod-S) denote the thick triangulated subcategory of K(Mod-S) consisting of T -acyclic complexes. The relative derived category of Mod-S with respect to T , denoted by DT (Mod-S), is defined to be the Verdier quotient K(Mod-S)/KT -ac (Mod-S), see 3.11. ✩

This research was in part supported by a grant from IPM (No. 93130216, 96130063).

* Corresponding author. E-mail addresses: [email protected], [email protected] (J. Asadollahi), [email protected] (R. Hafezi), [email protected], [email protected] (R. Vahed). https://doi.org/10.1016/j.jpaa.2018.05.015 0022-4049/© 2018 Elsevier B.V. All rights reserved.

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At the first part of the paper, i.e. Section 3, following similar methods in Rickard’s famous paper [31], we give a generalization of his theorem on derived equivalences of module categories over rings [31, Theorem 6.4] to functor categories, by showing, Theorem 3.14 below, that there is an equivalence of triangulated categories D(Mod-T )  DT (Mod-S). Moreover, if T has an additional property that thick(T ) = Kb (prj-S), then this equivalence translates to the following equivalence of triangulated categories D(Mod-T )  D(Mod-S). It will be shown that this later equivalence restricts to the bounded above and also bounded levels as well as to the homotopy category of projective objects, see Theorem 3.15. This theorem provides a sufficient condition for derived equivalences of functor categories without flatness assumption on the categories involved in the derived equivalence and so generalizes a result of Keller [18, 9.2, Corollary]. Note that the above triangle equivalences, at least on the levels of unbounded derived categories of functor categories and bounded homotopy categories of perfect complexes can be concluded also from [11, Theorem 7.5] and [32, Theorem 5.1.1], see Remark 3.16. The second part of the paper, i.e. Sections 4, contain applications. Let us review them briefly. In the first subsection of Section 4, we study a somehow converse of a result of Neeman. Neeman [27] proves that if A is a left coherent ring, then K(Prj-A), the homotopy category of projective (right) A-modules, is compactly generated and is an infinite completion of Db (mod-Aop )op . This in turn implies that if A and B are two right and left coherent rings such that K(Prj-A)  K(Prj-B), then A and B are derived equivalent, i.e. Db (Mod-A)  Db (Mod-B). We prove, Theorem 4.1.2, that the converse of this fact for left coherent rings A and B holds true, that is, if A and B are derived equivalent, then their homotopy categories K(Prj-A) and K(Prj-B) are also equivalent. Furthermore, this equivalence restricts to their full triangulated subcategories Ktac (Prj-A) and Ktac (Prj-B) consisting of totally acyclic complexes of projectives. In the second subsection of Section 4, we study the connection between the classical notion of derived equivalences with the notion of Gorenstein derived equivalences, introduced and studied in [14]. As a corollary, it will be shown that if Λ and Λ are coherent rings that are derived equivalent, then they are Gorenstein derived equivalent, i.e. there exists an equivalence between their Gorenstein derived categories DbGp (Mod-Λ) and DbGp (Mod-Λ ), Corollary 4.2.8. Recently, infinitely generated tilting modules over arbitrary rings have received considerable attention. In this connection, the notion of good tilting modules is introduced. We apply our results to show in Theorem 4.3.2 that every good n-tilting module TA provide an equivalence between the derived category D(Mod-A) and the relative derived category DT (Mod-EndA (T )op ). We refer the reader to Subsection 4.3 for definition of good n-tilting modules and their properties. 2. Preliminaries In general A denotes an associative ring with identity. We let Mod-A denote the category of all right A-modules. We also, consider the following full subcategories of Mod-A. Prj-A = projective (right) A-modules, mod-A = finitely presented A-modules, prj-A = finitely generated projective A-modules. 2.1. Let C be an additive category. We denote by C(C) the category of complexes in C. We grade complexes cohomologically, so every complex X  in C(C) is of the form

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n−1

3

n

· · · −→ X n−1 −∂−−−→ X n −∂−→ X n+1 −→ · · · . Let X  = (X i , ∂ i ) be a complex in C(C) and n, m be integers. We define the following truncations of X  :


n

n+1

: 0 −→ X n −∂−→ X n+1 −∂−−−→ X n+2 −→ · · ·

X  m = : · · · −→ X m−2 −∂−−−→ X m−1 −∂−−−→ X m −→ 0 m−2

m−1

We denote the homotopy category of C by K(C); the objects are complexes and morphisms are the homotopy classes of morphisms of complexes. The full subcategory of K(C) consisting of all bounded above, resp. bounded, complexes is denoted by K− (C), resp. Kb (C). Moreover, we denote by K−,b (C), the full subcategory of K− (A) formed by all complexes X  such that there is an integer n = n(X  ) with Hi (X  ) = 0, for all i ≤ n. Let A be an abelian category. The derived category of A will be denoted by D(A). Also, D− (A), resp. Db (A), denotes the full subcategory of D(A) formed by all homologically bounded above, resp. homologically bounded, complexes. 2.2. Total acyclicity Let X be an additive category. A complex X  in C(X ) is called X -totally acyclic if for every object Y ∈ X , the induced complexes HomX (X  , Y ) and HomX (Y, X  ) of abelian groups are acyclic. Let A be an abelian category having enough projective, resp. injective, objects and X = Prj-A, resp. X = Inj-A, be the class of projectives, resp. injectives. In this case, an X -totally acyclic complex is called totally acyclic complex of projectives, resp. totally acyclic complex of injectives. An object G in A is called Gorenstein projective, resp. Gorenstein injective, if G is a syzygy of a totally acyclic complex of projectives, resp. totally acyclic complex of injectives. We denote the class of all Gorenstein projective, resp. Gorenstein injective, objects in A by GP-A, resp. GI-A. In case A = Mod-A, we abbreviate the notations to GP-A and GI-A. We set Gp-A = GP-A ∩ mod-A and Gi-A = GI-A ∩ mod-A. 2.3. Localizing and thick subcategories Let D be a triangulated category with coproducts. A triangulated subcategory T of D is called thick if it is closed under direct summands and is called localizing if it is closed under coproducts in D. Let L be a class of objects of D. The smallest full triangulated thick subcategory of D containing L is denoted by thick(L) and the smallest full localizing subcategory of D containing L is denoted by LocL. Construction 2.4. For two classes L and H of objects of D, we let L ∗ H denotes the class of objects X occurring in an exact triangle Y → X → Z  in D, with Y ∈ L and Z ∈ H. Take L to be the class of all X[i] with X ∈ L and i ∈ Z. Define full subcategories L n , for n > 0, inductively as follows. • L 1 is the subcategory of D consisting of all direct summands of objects of L. • L n , for n > 1, is the full subcategory of D formed by all direct summands of objects of Ln , where Ln = L i ∗ L j with i, j < n. It can be easily checked that thick(L) =

 n∈N

L n , see [19, Lemma 3.3].

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 An object X of D is called compact if for every set {Yj }j∈J of objects of D, every map X → j∈J Yj factors through a finite coproduct. The full triangulated subcategory of D consisting of all compact objects will be denoted by Dc . We say that L generates D provided an object X of D is zero if for all L ∈ L, HomD (L[i], X) = 0, for all i ∈ Z. A triangulated category D is called compactly generated, if there is a set of compact objects generating D. Remark 2.5. There is another definition for generating sets of triangulated categories in terms of localizing subcategories. Based on this definition L generates D if the only localizing subcategory which contains the objects of L is D itself, where as above D is a triangulated category with coproducts and L is a set of objects of D. However, Lemma 2.2.1 of [32] shows that if L is a set of compact objects, then these two definitions are equivalent. Hence D is compactly generated if there is a set L of compact objects such that LocL = D. 2.6. Functor category Functor categories were introduced in representation theory of algebras by Auslander [3,4]. He used this kind of categories to classify Artin algebras of finite representation type [3] as well as to prove the first Brauer–Thrall conjecture [5]. Let S be a small pre-additive category, i.e. morphisms sets are abelian groups and the composition of morphisms are bilinear. Note that such S is called a ring with many objects in [11, §7] and also a ringoid in [32, §5]. Let Mod-S denote the category of additive contravariant functors from S to the category of abelian groups. Mod-S is called the category of modules over S, or the category of (right) S-modules. It is known that Mod-S is an abelian category with arbitrary coproducts. For an object S ∈ S, one may apply Yoneda lemma to show that the representable functor HomS (−, S) is a projective object in Mod-S. Moreover, for a functor F ∈ Mod-S, there is an epimorphism  i HomS (−, Si ) −→ F −→ 0, where Si runs through isomorphism classes of objects in S. So the abelian category Mod-S has enough projective objects. The full subcategory of Mod-S consisting of all projective objects will be denoted by Prj-S. An object F of Mod-S is called finitely presented if there exists an exact sequence  ⊕ni=1 HomS (−, Si ) −→ ⊕m j=1 HomS (−, Sj ) −→ F −→ 0,

for some integers n and m and objects Si and Sj of S. The category of all finitely presented S-modules is denoted by mod-S. Note that a finitely presented S-module F is projective if and only if it is isomorphic to a direct summand of a finite direct sum of representable functors. We denote by prj-S the full subcategory of mod-S consisting of all finitely presented projective functors. Let S be a small additive category in which every idempotent splits. Then, in this case, every finitely presented projective S-module is of the form HomS (−, S), for some S ∈ S. 3. Derived equivalences of functor categories In this section we provide a sufficient condition for derived equivalences of functor categories. Then we present a version of Rickard’s Theorem in terms of functor categories. The results in this section are relatively standard and most of the techniques, at least up to Proposition 3.5 below, are similar to their counterparts in [31]. Moreover, it should be emphasized that although the core could be obtained in a systematic way using the differential graded techniques of [11,18,32], we choose the following more elementary approach.

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3.1. The following definition is a generalization of the notion of a partial tilting complex, see also [18]. Let S be a small pre-additive category. A complex X over Mod-S is called perfect if it is quasi-isomorphic to a bounded complex of finitely presented projective S-modules. Let T be a set of perfect complexes over Mod-S satisfying the following properties: 



(P1) HomK(Mod-S) (T  , T  [i]) = 0, for all i = 0 and all T  , T  ∈ T ; i (P2) There exists a fixed integer n such that for each T  ∈ T , T  = 0, for |i| > n. Note that since every object of T is perfect, Condition (P1) implies that HomK(Mod-S) (T  , (⊕i∈I T  i )[n]) = 0, where T  ∈ T , I is an index set and T  i ∈ T for any i ∈ I and n = 0. Following a similar idea as in [31, p. 438], let KK(Mod-S) := K(K(Mod-S)) denote the homotopy category of K(Mod-S). Thus every object of KK(Mod-S) is a complex of complexes over Mod-S together with a sequence of homotopy classes of maps between them such that composing every two consecutive maps is null homotopic. For every object X in KK(Mod-S), we have: (a) a bigraded object X ∗∗ of Mod-S; (b) a graded endomorphism d of degree (1, 0), obtained from the differentials in the complexes over Mod-S; (c) a graded endomorphism δ of degree (0, 1), obtained from the original complex. Setup 3.2. Throughout the paper, unless otherwise specified, S denotes a small pre-additive category and T is a set of perfect complexes in K(Mod-S) satisfying the properties defined at 3.1. Also, let T denote a class of objects of KK(Mod-S) consisting of all stalk complexes T with T in degree zero, where T ∈ T . Note that for every complex T in T, there is the following isomorphism HomKK(Mod-S) (T, ⊕i Ti ) ∼ = ⊕i HomKK(Mod-S) (T, Ti ), where Ti ∈ T. Let KKT (Mod-S) be the smallest full triangulated subcategory of KK(Mod-S) that contains T and is closed under coproducts. Lemma 3.3. Let X be a complex in KKT (Mod-S). Then for every j, (X∗j , d) is a direct summand of a direct sum of objects of T . Proof. Let K(Add-T ) be the triangulated subcategory of KK(Mod-S) consisting of all complexes with terms in Add-T . Thus, if X is a complex in K(Add-T ), then for every j, (X∗j , d) is a direct summand of a direct sum of objects of T . Moreover, it is clear that K(Add-T ) is a triangulated subcategory of KK(Mod-S) which is closed under coproducts. Hence, K(Add-T ) is a localizing subcategory of KK(Mod-S) that contains T. Therefore, LocT is contained in K(Add-T ) and this implies the result. 2 One can use Lemma 3.3 together with properties (P1) and (P2) and follow, step by step, the same argument as in [31, Sec. 2] to construct a functor Φ : KKT (Mod-S) −→ K(Mod-S).

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The method that is used in [31, Proposition 2.11] can carry over verbatim to show the following proposition. Proposition 3.4. (See [31, Proposition 2.11].) The functor Φ : KKT (Mod-S) −→ K(Mod-S) is a triangle functor that preserves coproducts. Sketch of proof. Let G(S) be the category that is defined by Rickard as follows. An object is a system {X ∗∗ , di | i = 0, 1, · · · }, where X ∗∗ is a bigraded object of Mod-S satisfying the conditions as an object in KKT (Mod-S) such that X i∗ = 0 for large i, and for every i, di is a graded endomorphism of X ∗∗ of degree (1 − i, i) and for every n, d0 dn + d1 dn−1 + · · · + dn d0 = 0. The morphisms between two systems {X ∗∗ , di } and {Y ∗∗ , di } are collections {αi | i = 0, 1, · · · } of graded maps X ∗∗ −→ Y ∗∗ of degrees (−i, i) satisfying α0 dn + α1 dn−1 + · · · αn d0 = d0 αn + d1 αn−1 + · · · + dn α0 , for each n. Now we define a functor from KKT (Mod-S) to G(S) as follows. Let X be an object of KKT (Mod-S) and consider the bigraded object X ∗∗ of K(Mod-S), observe that Condition (P2) guarantees that X i∗ = 0 for large i. Hence to construct the desired functor, we just need to define the graded endomorphism di . We choose d0 = d and d1 = (−1)i+j δ : X i,j −→ X i,j+1 . Then other di ’s are defined inductively by using the same method as [31, Lemma 2.3]. Condition (P1) together with a similar argument as in the proof of Propositions 2.6 and 2.7 of [31] works to show that for any map α : X −→ Y in KKT (Mod-S), we can assign a morphism {αi } : X ∗∗ −→ Y ∗∗ of G(S). Now, we can define a triangulated functor Φ : KKT (Mod-S) −→ K(Mod-S) by going first to G(S) and then taking total complex. Just note that if {X ∗∗ , di | i = 0, 1, · · · } is an object  in G(S), then the sum di is well-defined. The method that is used in [31, Proposition 2.11] can carry over verbatim to show that Φ is in fact a triangulated functor. Also, definition of Φ implies that it preserves coproducts. 2 In the following, we plan to prove that the functor Φ defined above is fully faithful. Let KT (Mod-S) = LocT be the smallest full localizing subcategory of K(Mod-S) containing T . Proposition 3.5. (i) KKT (Mod-S) is compactly generated with T as a compact generating set. (ii) KT (Mod-S) is compactly generated with T as a compact generating set. Proof. (i) Let (−, T ) denote the class of all representable functors HomK(Mod-S) (−, T  ), where T  runs through objects of T . In view of Yoneda lemma, there is a bijection between T and (−, T ). This bijection

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can be extended to the equivalence ψ : Add-T −→ Prj-T . Indeed, for two families {T  i }i∈I and {T  j }j∈J of objects of T , there exist the following isomorphisms HomK(Mod-S) (⊕i∈I T  i , ⊕j∈J T  j )

 ∼ = i∈I HomK(Mod-S) (T  i , ⊕j∈J T  j )  ∼ = i∈I ⊕j∈J HomK(Mod-S) (T  i , T  j )  ∼ = i∈I ⊕j∈J HomMod-T (Hom(−, T  i ), Hom(−, T  j ))  ∼ HomMod-T (Hom(−, T  i ), ⊕j∈J Hom(−, T  j )) = i∈I

∼ = HomMod-T (⊕i∈I Hom(−, T  i ), ⊕j∈J Hom(−, T  j )) ∼ = HomMod-T (Hom(−, ⊕i∈I T  i ), Hom(−, ⊕j∈J T  j )). Now, since every object of Add-T , resp. Prj-T , is a direct summand of some objects of the form ⊕Ti , resp. ⊕i Hom(−, Ti ), we infer that ψ is fully faithful. Moreover, it follows directly from the definition that ψ : Add-T −→ Prj-T is dense. This equivalence can be extended naturally to the equivalence K(ψ) : K(Add-T ) −→ K(Prj-T ). By the proof of Lemma 3.3, LocT is contained in K(Add-T ). Moreover, the image of LocT under the functor K(ψ) is the smallest triangulated subcategory of K(Prj-T ) containing K(ψ)(T) and closed under coproducts. Since K(ψ)(T) = (−, T ), the image of LocT under the functor K(ψ) is Loc(−, T ) and so there is the following commutative diagram K(Add-T )

LocT

K(ψ)

K(ψ)|

K(Prj-T )

Loc(−, T ),

where rows are triangle equivalences. It is known that each object of (−, T ) is compact in K(Prj-T ). Thus, Remark 2.5 yields that Loc(−, T ) is compactly generated. Hence KKT (Mod-S) = LocT is generated by T. This completes the proof of (i). (ii) The result follows from the fact that every object of T is compact in K(Mod-S) together with Remark 2.5. 2 Theorem 3.5 of [31] states the following result in module case. Here, we prove this for functor categories, using a completely different approach. Proposition 3.6. The triangulated functor Φ : KKT (Mod-S) −→ K(Mod-S) is full and faithful. Moreover, it induces the equivalence Φ : KKT (Mod-S) −→ KT (Mod-S), of triangulated categories. Proof. Observe that by definition of Φ, if T is a talk complex with T in degree zero, where T ∈ T , then Φ(T) = T . Hence, the image of the triangulated functor Φ is contained in KT (Mod-S). In view of Proposition 3.5, KKT (Mod-S) and KT (Mod-S) are compactly generated with compact generating sets T and T , respectively. So, by [22, Proposition 6], it suffices to show that the restriction of Φ to compact objects is an equivalence. By Lemma 2.2 of [24], KKT (Mod-S)c = thick(T) and KT (Mod-S)c = thick(T ).  According to Construction 2.4, thick(T ) = n∈N T n . Now, we show that Φ| : thick(T) −→ thick(T ) is full and faithful, using induction on n.

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First assume that n = 1. Property (P1) together with compactness of objects of T implies that if X and Y belong to T 1 , then HomKKT (Mod-S) (X, Y) ∼ = HomKT (Mod-S) (Φ(X), Φ(Y)). Now let n > 1 and X and Y belong to T n . Hence there exist triangles X → X → X , Y → Y → Y , where X ∈ T m , Y ∈ T m , X ∈ T l and Y ∈ T l such that m, m , l, l < n. So, by induction hypothesis, we have the following isomorphism HomKKT (Mod-S) (X, Y) ∼ = HomKT (Mod-S) (Φ(X), Φ(Y)). Consequently, Φ| : KKT (Mod-S)c −→ KT (Mod-S)c is full and faithful. Clearly, Φ| is also dense and so is an equivalence. 2 b Denote by K− T (Mod-S), resp. KT (Mod-S), the full triangulated subcategory of KT (Mod-S) consisting of all bounded above, resp. bounded, complexes. Let KT−,T b (Mod-S) denote the full triangulated subcat  egory of K− T (Mod-S) consisting of all complexes X , in which there is an integer n = n(X ) such that −    HomKT (Mod-S) (T , X [i]) = 0 for all i < n and every T ∈ T . Also, we denote by KKT (Mod-S) the full −,Tb subcategory of KKT (Mod-S) formed by all bounded above complexes. Moreover, KKT (Mod-S) denotes − the full triangulated subcategory of KKT (Mod-S) formed by all complexes X in which there is an integer n = n(X) such that HomKKT (Mod-S) (T, X[n]) = 0 for all i < n and every T ∈ T.

Let D be a triangulated category with direct sums. Let L be a set of objects of D. Then ⊕L denotes the set of all direct sums of objects of L. In the following we show that the triangle equivalence Φ defined above can be restricted to various subcategories of KKT (Mod-S) and KT (Mod-S). Proposition 3.7. There is a commutative diagram Φ

KKT (Mod-S)

∼

KT (Mod-S)

KK− T (Mod-S)

∼

K− T (Mod-S)

−,Tb KKT (Mod-S)

∼

KT−,T b (Mod-S)

thick(⊕T)

thick(T)

∼

∼

thick(⊕T )

thick(T )

of triangulated categories whose rows are triangle equivalences.

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For the proof, we need the following three lemmas. Lemma 3.8. (i) Let X be a complex in KKT (Mod-S). Then X lies in KK− T (Mod-S), up to isomorphism, if and only if there exists an integer N > 0 such that HomKK(Mod-S) (T[−n], X) = 0, for all n ≥ N and every complex T in T. (ii) Let X  be a complex in KT (Mod-S). Then X  lies in K− T (Mod-S), up to isomorphism, if and only if there exists an integer N > 0 such that HomK(Mod-S) (T  [−n], X  ) = 0, for all n ≥ N and every complex T  in T . Proof. These statements follow directly from the facts that KKT (Mod-S) = LocT and KT (Mod-S) = LocT , respectively. 2 Lemma 3.9. (See [31, Proposition 6.1].) −,Tb (i) An object X ∈ KK− (Mod-S), up to isomorphism, if and only if for each T (Mod-S) lies in KKT − complex Y ∈ KKT (Mod-S) there exists an integer N such that HomKKT (Mod-S) (Y, X[n]) = 0, for all n < N. −,T b (Mod-S), up to isomorphism, if and only if for each complex (ii) An object X  ∈ K− T (Mod-S) lies in KT −  Y ∈ KT (Mod-S) there exists an integer N such that HomKT (Mod-S) (Y  , X  [n]) = 0, for all n < N .

Proof. (i) First note that a modification of the proof of [31, Proposition 6.1] works to prove that, for a small pre-additive category A, a complex X in K− (Prj-A) lies in K−,b (Prj-A), up to isomorphism, if and only if for every complex Y ∈ K− (Prj-A) there is a natural number N such that Hom(Y, X[n]) = 0, for all n < N . Now, the equivalence K(ψ) : LocT −→ Loc(−, T ) of triangulated categories yields the result. (ii) Let X  be a complex in KT−,T b (Mod-S). Hence there is an integer n such that Hom(T  , X  [i]) = 0 for all T  ∈ T and i < n. On the other hand, the triangle equivalence Φ : KKT (Mod-S) −→ KT (Mod-S) together with Lemma 3.8  implies that there exists a complex X ∈ KK− T (Mod-S) such that Φ(X) = X . Moreover, for every complex T ∈ T, there exist the following isomorphisms HomKK(Mod-S) (T, X[i])

∼ = HomK(Mod-S) (Φ(T), Φ(X)) ∼ = HomK(Mod-S) (Φ(T), X  [i]).

Since, Φ(T) lies in T , HomK(Mod-S) (Φ(T), X  [i]) = 0, for all i < n. This means that X belongs to −,Tb KKT (Mod-S).

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Now, let Y  be a complex in K− T (Mod-S). Again the equivalence Φ and Lemma 3.8 implies that there is a complex Y in KK− (Mod-S) such that Φ(Y) = Y  and so we have the following isomorphisms T HomK(Mod-S) (Y  , X  [i])

∼ = HomK(Mod-S) (Φ(Y), Φ(X)[i]) ∼ = HomKK(Mod-S) (Y, X[i]).

It follows from part (i) that there is an integer N such that HomKK(Mod-S) (Y, X[i]) = 0, for all i < N . This in turn implies that HomK(Mod-S) (Y  , X  [i]) = 0 for all i < N . The same argument as above can be applied to show the converse. 2 Lemma 3.10. (See [31, Proposition 6.2].) −,Tb (Mod-S) lies in thick(⊕T), up to isomorphism, if and only if for each complex (i) An object X ∈ KKT −,Tb Y ∈ KKT (Mod-S) there is an integer N such that

HomKKT (Mod-S) (X, Y[n]) = 0, for all n > N . (ii) An object X  ∈ KT−,T b (Mod-S) lies in thick(⊕T ), up to isomorphism, if and only if for each complex b Y  ∈ K−,T (Mod-S) there is an integer N such that T HomKT (Mod-S) (X  , Y  [n]) = 0, for all n > N . Proof. (i) Let A be a small pre-additive category. One should apply an argument similar to [31, Proposition 6.2] verbatim to show that a complex X in K−,b (Prj-A) lies in Kb (Prj-A), up to isomorphism, if and only if for every complex Y ∈ K−,b (Prj-A), Hom(X, Y [n]) = 0, for large n. Now, the equivalence K(ψ) : LocT −→ Loc(−, T ) implies the desired characterization and completes the proof. The statement (ii) can be obtained from the same argument as in the proof of Lemma 3.9(ii) in conjunction with (i). 2 Proof of Proposition 3.7. First observe that every triangle equivalence induces an equivalence on the compact objects. Hence, in view of Proposition 3.5, the bottom row is an equivalence. The other induced equivalences are direct consequences of Lemmas 3.8, 3.9 and 3.10. 2 3.11. Relative derived category Let S be a small pre-additive category and X be a full subcategory of K(Mod-S). A complex Y  over Mod-S is called X -acyclic, if for every object X  of X and all i ∈ Z, HomK(Mod-S) (X  , Y  [i]) = 0. Let K∗X -ac (Mod-S) denote the full triangulated subcategory of K∗ (Mod-S) consisting of all X -acyclic complexes, where ∗ ∈ {blank, −, b}. Clearly, K∗X -ac (Mod-S) is a thick subcategory of K∗ (Mod-S). So we get the triangulated category D∗X (Mod-S) := K∗ (Mod-S)/K∗X -ac (Mod-S)

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for ∗ ∈ {blank, −, b}, which is called the relative derived category of Mod-S with respect to X . A similar proof as in [14, Proposition 2.7] works to show that D− X (Mod-S) is a triangulated subcategory of DX (Mod-S), and DbX (Mod-S) is a triangulated subcategory of D− X (Mod-S), and so of DX (Mod-S). Recall that if D is a triangulated subcategory of D, then the left orthogonal of D in D is defined by ⊥

D = {X ∈ D | HomD (X, Y ) = 0, for all Y ∈ D }.

The right orthogonal is defined similarly. The proof of the following lemma is a standard application of the Bousfield localization techniques from [26]. A nice exposition is also given in Section 4 of [20]. Moreover, note that this theory had been studied before in the setting of algebraic triangulated categories, see e.g. Corollary 3.2 of [28]. Lemma 3.12. There is an equivalence DT (Mod-S)  KT (Mod-S) of triangulated categories. Proof. Consider the following sequence of functors ι

Q

KT (Mod-S) → K(Mod-S) → DT (Mod-S), where ι is the inclusion and Q is the canonical functor. Since KT (Mod-S) = LocT , KT (Mod-S) ⊆ ⊥ KT -ac (Mod-S). Hence, by Lemma 9.1.5 of [26], the composition functor Q ◦ ι : KT (Mod-S) −→ DT (Mod-S) is full and faithful. So to complete the proof, it remains to show that Q ◦ ι is dense. Since, by Proposition 3.5 (i), KT (Mod-S) = LocT is compactly generated, Theorem 4.1 of [25] implies that the inclusion ι

KT (Mod-S) → K(Mod-S) has a right adjoint. So, for every complex X  in K(Mod-S), there is a triangle 



X  → X  → X  , 



in K(Mod-S), where X  ∈ KT (Mod-S) and X  ∈ (KT (Mod-S))⊥ . Since T is contained in KT (Mod-S),  X  is a T -acyclic complex. Therefore, if we consider the above triangle in DT (Mod-S) under the canonical  functor Q, we get that X  is isomorphic to X  . This means that Q ◦ ι is dense. 2 Remark 3.13. In view of the proof of Proposition 3.5 there is an equivalence K(ψ) : LocT −→ Loc(−, T ). It is known that D(Mod-T ) is compactly generated with (−, T ) as a compact generating set, this is a

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consequence of results of [21]. That is Loc(−, T )  D(Mod-T ) as triangulated categories. Consequently, we have the following equivalence of triangulated categories ∼

∼

D(Mod-T ) −→ Loc(−, T ) −→ LocT = KKT (Mod-S). Theorem 3.14. As above, let T be a set of perfect complexes over Mod-S satisfying Conditions (P1) and (P2) of 3.1. Then there is the following commutative diagram of triangulated categories with equivalence rows D(Mod-T ) D− (Mod-T ) Db (Mod-T ) Kb (Prj-T ) Kb (prj-T )

∼

∼

∼

∼

∼

DT (Mod-S) D− T (Mod-S) DbT (Mod-S) thick(⊕T )

thick(T )

Proof. By Remark 3.13 there is an equivalence D(Mod-T )  KKT (Mod-S). Moreover, Proposition 3.6 implies that KKT (Mod-S)  KT (Mod-S). Hence, there exists the first equivalence thanks to Lemma 3.12. Q◦ι| − For the second equivalence, consider the restriction functor K− T (Mod-S) −−−→ DT (Mod-S). Since Q ◦ ι is full and faithful, we just need to prove that Q ◦ ι| is dense as well. Let X be a complex in K− (Mod-S). As in the proof of Lemma 3.12, there exists a triangle 



X  → X  → X  , 



where X  ∈ KT (Mod-S) and X  is a T -acyclic complex. Since X  ∈ K− (Mod-S), for each complex T  in thick(T ), there is an integer N = N (T  ) such that  HomK(Mod-S) (T  [−n], X  ) = 0, for all n ≥ N . Also, HomK(Mod-S) (T  [n], X  ) = 0, for all n ∈ Z and   all T ∈ thick(T ). Thus, for every complex T ∈ thick(T ) there is an integer N = N (T  ) such that   HomK(Mod-S) (T  [−n], X  ) = 0, for all n ≥ N . Hence, by Lemma 3.8, X  belongs to K− T (Mod-S). The im−  ∼  (Mod-S) implies that X in D (Mod-S) and so Q ◦ι| : K− age of the above triangle in D− X = T T T (Mod-S) −→ ∼ − DT (Mod-S) is an equivalence. On the other hand, Lemma 3.8 implies that the equivalence D(Mod-T ) −→ ∼ KKT (Mod-S) introduced in Remark 3.13, restricts to the equivalence D− (Mod-T ) −→ KK− T (Mod-S). − − So Proposition 3.7 yields the desired equivalence D (Mod-T ) −→ DT (Mod-S) that commutes the first square. The same argument as above together with Lemma 3.9 implies the existence of the equivalence ∼ b D (Mod-T ) −→ DbT (Mod-S) making the second square commutative. Similarly, the characterization of thick(⊕T ), Lemma 3.10, in conjunction with Proposition 3.7 yield the equivalence Kb (Prj-T ) −→ thick(⊕T ) making the diagram commutative. The last equivalence follows from the fact that every equivalence between triangulated categories can be restricted to the equivalence between their compact objects. 2

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Theorem 3.15. As in 3.1, let T be a set of perfect complexes over Mod-S satisfying properties (P1) and (P2) and thick(T ) = Kb (prj-S). Then there exists the following commutative diagram D(Mod-T ) D− (Mod-T ) Db (Mod-T ) Kb (Prj-T ) Kb (prj-T )

∼

∼

∼

D(Mod-S) D− (Mod-S) Db (Mod-S)

∼

∼

Kb (Prj-S) Kb (prj-S)

in which rows are equivalences of triangulated categories. Proof. It is easy to check that if thick(T ) = Kb (prj-S), then a complex in K(Mod-S) is acyclic if and only if it is T -acyclic. Indeed, since T contains all finitely presented projective S-modules, every T -acyclic complex is acyclic. Moreover, T is a subcategory of Kb (prj-S). So an induction argument on the length of complexes in T shows that every acyclic complex is T -acyclic. Therefore, definition of the relative derived category implies that DT (Mod-S) coincides with D(Mod-S). Hence, in view of Theorem 3.14, we have the desired diagram. Just note that the same argument as in the proof of [31, Proposition 6.2] implies the third square, while the last equivalence follows from the fact that every equivalence between triangulated categories induces an equivalence between their compact objects. 2 Remark 3.16. Let S be a small pre-additive category. A subcategory R of Kb (prj-S) is called a tilting   subcategory if it satisfies (P1), i.e. HomK(Mod-S) (X  , X  [i]) = 0, for all i = 0 and all X  , X  ∈ R, and HomS (−, X) ∈ thick(R) for all X ∈ S. A tilting subcategory is called a set of tiltors in [11,32]. It is proved in [11, Theorem 7.5] and [32, Theorem 5.1.1] that if there is a set of tiltors T in K(Mod-S), then there exist the above triangle equivalences on the level of unbounded derived categories of functor categories and bounded homotopy categories of perfect complexes. Let R be a commutative ring. A small pre-additive category S is called R-flat, if S(x, y) is flat R-module, for every x, y ∈ S. Keller [18, 9.2, Corollary] proved that two R-flat categories S and S  are derived equivalent, i.e. D(Mod-S)  D(Mod-S  ), if and only if there exists a tilting subcategory R of Kb (prj-S) such that R is equivalent to S  . Therefore, in the above theorem we provide a sufficient condition for derived equivalences of functor categories without flatness assumption on the categories involved in the derived equivalence. Remark 3.17. Our proofs show that the results of this section can be extended to a skeletally small preadditive category S. 4. Applications This section contains some interesting applications of the equivalences of Section 3. It is divided to three subsections.

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4.1. Derived equivalences versus equivalences of homotopy categories Let A and B be two rings. Then A and B are called derived equivalent if there exists a triangle equivalence D (Mod-A)  Db (Mod-B). It is proved by Rickard [31, Sec. 8] that if A and B are right coherent, then A and B are derived equivalent provided there exists a triangle equivalence Db (mod-A)  Db (mod-B). Let A be a left coherent ring. It is known, by [27, Theorem 1.1], that K(Prj-A) is compactly generated and c K (Prj-A)  Db (mod-Aop )op . Moreover, every equivalence between triangulated categories can be restricted to an equivalence between their compact objects. Hence, if A and B are two right and left coherent rings such that K(Prj-A)  K(Prj-B), then A and B are derived equivalent. This subsection deals with the converse of this fact. b

4.1.1. Let A be an Artin k-algebra, where k is a commutative artinian ring. Then A is called Gorenstein if id A A < ∞ and id AA < ∞. For a class X of A-modules, the left and right orthogonals of X are defined as follows ⊥

Y := {M ∈ Mod-A | Ext1A (M, Y ) = 0, for all Y ∈ Y}

and X ⊥ := {M ∈ Mod-A | Ext1A (X, M ) = 0, for all X ∈ X }. The Artin algebra A is called virtually Gorenstein if (GP-A)⊥ = ⊥ (GI-A). Theorem 4.1.2. Let A and B be two left coherent rings that are derived equivalent. Then there is a commutative diagram α1

K(Prj-B) α2

K(prj-B)

Ktac (Prj-B) Ktac (prj-B)

β2

K(Prj-A)

K(prj-A) β1

Ktac (Prj-A)

Ktac (prj-A)

in which α1 and β1 are triangulated equivalences. Moreover, if A and B are virtually Gorenstein algebras, then β2 is also an equivalence of triangulated categories. Proof. By Theorem 4.6 of [31] there exists a tilting complex T  in Kb (prj-A) such that EndK(prj-A) (T  ) ∼ = B. In [31, Section 2], Rickard constructed a triangle functor K− (Sum-T  ) −→ K− (Prj-A). One should use Rickard’s construction to obtain a triangle functor K(Sum-T  ) −→ K(Prj-A). Indeed, let (X∗∗ , d, ∂) be an object of K(Sum-T  ). Then X ∗∗ is a bigraded object of Prj-A such that X i∗ = 0 for large i. Moreover, for each j ∈ Z, (X∗j , d) is a direct sum of T  . So by using the same method as in [31, Section 2] we obtain a bigraded complex X∗∗ together with a graded endomorphism di of X∗∗ of degree (1 − i, i) such that for  every n, d0 dn + d1 dn−1 + · · · + dn d0 = 0. Next take the total complex of X∗∗ with differential di to get an unbounded complex of projective Λ-modules.

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Now, consider the triangulated category K(Add-T  ). Each object of K(Add-T  ) is a complex of complexes in Add-T  together with a sequence of homotopy classes of maps between them such that composing every ∼ two consecutive maps is null homotopic. There is a triangle equivalence K(Sum-T  ) −→ K(Add-T  ) induced by the inclusion Sum-T  → Add-T  . Therefore, there exists a triangle functor Φ : K(Add-T  ) −→ K(Prj-A). On the other hand, the functor HomK(prj-A) (T  , −) provides an equivalence K(Add-T  )  K(Prj-B). Hence K(Add-T  ) is a compactly generated triangulated category and since Φ preserves coproducts, there is the following triangle functor Φ| : K(Add-T  )c −→ K(Prj-A)c . Moreover, by [27, Proposition 7.12], a complex in K(Add-T  ) is compact if and only if it is isomorphic to a complex X satisfying (i) X is a complex with terms in add-T  , (ii) X n = 0, for n  0, (iii) HomK(Add-T  ) (X, T  [n]) = 0, for n  0. ∗

It is known that if T  is a tilting complex over a ring A, then T  := HomA (T  , A) is a tilting complex over ∗ Aop . Similarly, we have a triangulated functor Φ∗ : K(Add-T  ) −→ K(Prj-Aop ) that preserves coproducts. Furthermore, the above equivalence K(Add-T  )  K(Prj-A) together with Proposition 6.1 of [31] yields that Φ∗ restricts to the following triangle equivalence Φ∗ : K−,T

∗

b

∗

(add-T  ) −→ K−,b (prj-Aop ).

∗ ∗  Let the contravariant functor Hom(−, A) : K(Add-T  )c −→ K−,T b (add-T  ) map any complex ∗ ∗ (X∗∗ , d, δ) of K(Add-T )c to the complex ((X∗∗ , A), (d, A), (δ, A)) of K−,T b (add-T  ). Then there is the following commutative diagram of triangulated categories

Φ|

K(Add-T  )c

K(Prj-A)c

 Hom(−,A)

K−,T

∗

b

Hom(−,A) ∗

(add-T  )

Φ∗ |

K−,b (prj-Aop ).

Indeed, let (X∗∗ , d, ∂) be a complex in K(Add-T  )c . Then X satisfies the above characterization of compact ∗ ∗ objects in K(Add-T  ) and so the induced complex ((X∗∗ , d), (d, A), (δ, A)) belongs to K−,T b (add-T  ). By ∗ ∗∗ definition, the functor Φ maps a complex ((X , A), (d, A), (δ, A)) to a complex Y which is the total complex  ∗ of (X ∗∗ , A) with differential di , where d∗i is a graded endomorphism of (X ∗∗ , A) of degree (1 − i, i). Now, apply the functor Hom(−, A) on Y  to obtain a complex Hom(Y  , A). Since for each i, Y i = ⊕j (X j,i−j , A), the i-th term of Hom(Y, A) is of the form ⊕j ((X j,i−j , A), A). Moreover, X i,j ∈ prj-A, for every i, j ∈ Z and so ((X i,j , A), A) ∼ = X i,j . This fact shows that Hom(Y  , A) is isomorphic to a complex in K(Prj-A)c with  j,i−j ⊕j X as i-th degree and di as its differential. This means that Hom(Y  , A) ∼ = Φ(X). ∗  Since Φ is an equivalence, and Hom(−, A) and Hom(−, A) are dualities, Φ| must be an equivalence. Proposition 6 of [22] implies that Φ : K(Add-T  ) −→ K(Prj-A) is an equivalence. Consequently, there exists an equivalence ∼

α1 : K(Prj-B) −→ K(Prj-A).

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To prove that β1 = α1 | is an equivalence, it is enough to show that α1 maps any complex in Ktac (Prj-B) to a complex of Ktac (Prj-A) and moreover, show that β1 is dense. First observe that by Theorem 3.14, the equivalence α1 can be restricted to the triangle equivalence α1 | : Kb (Prj-B) −→ Kb (Prj-A). So, if P is a projective A-module, then there is a bounded complex Q of projective B-modules such that α1 (Q ) ∼ = P. Let X  be a complex in Ktac (Prj-B). Then for each P ∈ Prj-A and each i ∈ Z, there are the following isomorphisms HomK(Prj-A) (P, α1 (X  )[i])

∼ = HomK(Prj-A) (α1 (Q ), α1 (X  )[i]) ∼ = HomK(Prj-B) (Q , X  [i])

HomK(Prj-A) (α1 (X  )[i], P )

∼ = HomK(Prj-A) (α1 (X  )[i], α1 (Q )) ∼ = HomK(Prj-B) (X  [i], Q ),

and

where Q ∈ Kb (Prj-B). Since X  ∈ Ktac (Prj-B), for every Q ∈ Prj-B and all i ∈ Z HomK(Prj-B) (Q, X  [i]) = 0 = HomK(Prj-B) (X  [i], Q). So, using an induction argument on the length of Q one can deduce that HomK(Prj-A) (P , α1 (X  )[i]) = 0 = HomK(Prj-A) (α1 (X  )[i], P ), where P ∈ Prj-A and i ∈ Z. A similar argument as above implies that β1 is dense. Furthermore, it is clear that there exist the induced functors α2 and β2 that are full and faithful. In case A and B are virtually Gorenstein algebras, then, by [8, Theorem 8.2], Kctac (Prj-A)  Ktac (prj-A) and Kctac (Prj-B)  Ktac (prj-B). Therefore, β2 is an equivalence. 2 Remark 4.1.3. Benson and Krause [9] proved a similar statement as above for the homotopy category of injectives, using a different approach. Let A and B be noetherian algebra over a commutative ring k such that A and B are projective k-modules. Then, it is proved in [9, Proposition 8.1] that if A and B are derived equivalent, then there is a triangle equivalence K(Inj-A)  K(Inj-B). Remark 4.1.4. Let A and B be two right and left coherent rings that are derived equivalent. By [31, Propo∼ ∼ sition 9.1], there is an equivalence γ : Db (mod-Aop ) −→ Db (mod-B op ), and so γ op : Db (mod-Aop )op −→ Db (mod-B op )op is an equivalence. The equivalence α : K(Prj-A) −→ K(Prj-B), that is proved in Theorem 4.1.2, can be viewed as an extension of the equivalence γ op . In fact, by a result of Neeman [27, Theorem 1.1], K(Prj-A), resp. K(Prj-B), is compactly generated and Kc (Prj-A)  Db (mod-Aop )op , resp. Kc (Prj-B)  Db (mod-B op )op . Moreover, these maps fit into the following commutative diagram Kc (Prj-A)

α|



Db (mod-Aop )op

Kc (Prj-B) 

γ

op

Db (mod-B op )op .

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In particular, A and B are derived equivalent. The above theorem provides the converse of this fact for left coherent rings. Remark 4.1.5. Let GP-A, resp. Gp-A, denote the stable category of GP-A, resp. Gp-A, modulo the full subcategory Prj-A, resp. prj-A. In view of the known equivalences Ktac (Prj-A)  GP-A, β1 in Theorem 4.1.2 is, in fact, an equivalence GP-A  GP-B. So, it follows from Theorem 4.1.2 that if A and B are two left coherent rings that are derived equivalent, then there is a triangle equivalence GP-A  GP-B. This result has already been proved by Hu and Pan [15] for two arbitrary rings, but using a different approach. Remark 4.1.6. In [17] Jørgensen proves that if A is a commutative noetherian ring with a dualizing complex, then the inclusion functor Ktac (Prj-A) → K(Prj-A) admits a right adjoint. This result was generalized in [23] to commutative noetherian rings of finite Krull dimension. Moreover, it is easy to check that, for an arbitrary ring A, the existence of a right adjoint for the inclusion Ktac (Prj-A) → K(Prj-A) implies the existence of Gorenstein projective precovers over A, see [17]. On the other hand, Theorem 4.1.2 implies that if A and B are two rings that are derived equivalent, then the inclusion functor Ktac (Prj-A) → K(Prj-A) has a right adjoint if and only if the inclusion functor Ktac (Prj-B) → K(Prj-B) has a right adjoint. Hence, if A is a commutative noetherian ring of finite Krull dimension and B is a ring such that Db (Mod-A)  Db (Mod-B), then Gorenstein projective B-modules is a precovering class of Mod-B. 4.2. Derived equivalences versus Gorenstein derived equivalences In this subsection, we show that over right coherent rings, certain derived equivalences imply Gorenstein derived equivalences. Let us first recall briefly the definition of Gorenstein derived categories. Let A be a right coherent ring. A complex X  of finitely generated A-modules is called Gp-acyclic if for every G ∈ Gp-A, the induced complex HomA (G, X  ) is acyclic. We denote by KbGp-ac (mod-A) the class of all Gp-acyclic complexes in Kb (mod-A). The bounded Gorenstein derived category of mod-A, denoted by DbGp (mod-A), is the quotient category Kb (mod-A)/KbGp-ac (mod-A). The Gorenstein derived category was studied by Gao and Zhang [14]. Recently this category has been studied more in [1,2]. 4.2.1. A complex X  in D− (mod-A) is called of finite projective dimension if the functor HomD(mod-A) (X  [i], −) vanishes on mod-A, for i  0. The full subcategory of D− (mod-A) consisting of all complexes of finite projective dimension is denoted by Db (mod-A)fpd . It is known that there exists an equivalence ∼

Q : K− (prj-A) −→ D− (mod-A), where Q is the canonical functor. Moreover, this equivalence induces the equivalence ∼

Kb (prj-A) −→ Db (mod-A)fpd of triangulated categories. The singularity category of A, denoted by Dbsg (A), is then the Verdier quotient Db (mod-A)/Db (mod-A)fpd . Let F : Db (mod-A) −→ Db (mod-B) be a derived equivalence with the quasi-inverse G. Suppose that  T ∈ Kb (prj-A), resp. T  ∈ Kb (prj-B), is the tilting complex associated to F , resp. G. By [16, Lemma 2.1],  we assume that T  and T  are complexes of the form 

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T :

0 −→ T −n −→ T −n+1 −→ · · · −→ T −1 −→ T 0 −→ 0,



T :

0 −→ T  0 −→ T  1 −→ · · · −→ T  n−1 −→ T  n −→ 0.

We fix these notations towards the end of this subsection. We need the following lemmas. Lemma 4.2.2. ([16, Lemma 2.2]) Let X  be a bounded above and Y  be a bounded below complex over a ring A. If there is an integer m, such that X i is projective for all i > m and Y j = 0 for all j < m, then HomD(Mod-A) (X  , Y  ) ∼ = HomK(Mod-A) (X  , Y  ). Lemma 4.2.3. Let A and B be two right coherent rings that are derived equivalent. Let F : Db (mod-A) −→ Db (mod-B) be an equivalence with the quasi-inverse G. (i) Assume that X is a finitely generated Gorenstein projective B-module. Then G(X) is isomorphic in Db (mod-A) to a complex of the form −n −n+1 −1 0 T  X : 0 −→ TX −→ TX −→ · · · −→ TX −→ TX −→ 0 −n i where TX is Gorenstein projective and TX is projective, for −n + 1 ≤ i ≤ 0. (ii) Assume that X is a finitely generated Gorenstein projective A-module. Then F (X) is isomorphic in Db (mod-B) to a complex of the form 

T  X : 0 −→ T  X −→ T  X −→ · · · −→ T  X 0

1

n−1

−→ T  X −→ 0 n

where T  X is Gorenstein projective and T  X is projective, for 1 ≤ i ≤ n. 0

i

Proof. The result follows from Lemma 4.5 of [15] together with [15, Proposition 5.2(2)]. 2 Lemma 4.2.4. Using the same notations as in Lemma 4.2.3, for each pair X, Y ∈ Gp-B and i = 0, we have HomKb (mod-A) (T  X , T  Y [i]) = 0. Proof. The result can be obtained from a simple modification of the proof of [30, Lemma 3.7]. For the convenience of the reader, we include the sketch of the proof here. Lemma 4.2.2 implies that HomKb (mod-A) (T  X , T  Y [i]) = 0, for i < 0. For i > 0, consider the following triangles in Kb (mod-A) −n [n] −→ T¯ X [1], T¯ X −→ T  X −→ TX

T¯ Y −→ T  Y −→ TY−n [n] −→ T¯ Y [1], where T¯ X , resp. T¯ Y , denotes the complex <−n+1 T  X , resp. <−n+1 T  Y . The second triangle implies that Hi (F (T¯ Y )) = 0, for all i > 1. So F (T¯ Y ) is isomorphic in Db (mod-B) to a complex Q of the form · · · −→ Q−1 −→ Q0 −→ Q1 −→ 0. Now, apply the cohomological functors HomK(mod-A) (−, T¯ Y [i]) and HomD(mod-A) (−, T¯ Y [i]) on the first triangle and the homological functor HomK(mod-A) (T  X , −) on the second one. One can deduce that HomK(mod-A) (T  X , T  Y [i]) = 0 for i > 1.

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To prove that HomKb (mod-A) (TX , TY [1]) = 0, it is enough to show that the induced map HomKb (mod-A) (T  X , TY−n [n]) −→ HomKb (mod-A) (T  X , T¯ Y [1]) is surjective. Consider the commutative diagram Q

α

Y

∼ =

∼ =

∼ =

F (TY−n [n])

F (T  Y )

F (T¯ Y )

Q [1]

cone(α)

∼ =

F (T¯ Y [1]).

Apply Lemma 4.2.2 to get the following commutative diagram whose vertical maps are isomorphisms HomKb (mod-A) (T  X , TY−n [n])

HomKb (mod-A) (T  X , T¯ Y [1])

∼ =

∼ =

HomDb (mod-B) (X, Q [1]).

HomDb (mod-B) (X, cone(α)) So, it is enough to prove the following isomorphisms

HomDb (mod-B) (X, Q [1]) ∼ = HomKb (mod-B) (X, Q [1]), HomDb (mod-B) (X, Cone(α)) ∼ = HomKb (mod-B) (X, Cone(α)). The first isomorphism can be proved by induction on the length of Q , and the second one is obtained by applying the homological functors HomDb (mod-B) (X, −) and HomKb (mod-B) (X, −) on the triangle cone(α)0 −→ cone(α) −→ cone(α)−1 = −→ cone(α)0 [1] in Kb (mod-B). Note that in the above triangle, cone(α)0 denotes the stalk complex with the zeroth term of the complex cone(α) in degree zero. 2 Lemma 4.2.5. Let A be a ring. Then there exists an equivalence Kb (Gp-A)  Kb (prj-(Gp-A)) of triangulated categories. Proof. Consider the functor ϕ : Gp-A −→ prj-Mod(Gp-A) given by ϕ(X) = HomA (−, X). Every object of prj-Mod(Gp-A) is of the form HomA (−, X) for some X ∈ Gp-A. So, by using Yoneda lemma one can easily deduce that ϕ is an equivalence. This equivalence can be extended, in a natural way, to the desired equivalence ∼

ϕ¯ : Kb (Gp-A) −→ Kb (prj-(Gp-A)) of triangulated categories. 2

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Theorem 4.2.6. Let A and B be two right coherent rings that are derived equivalent. Then there is an equivalence D(Mod-Gp-A)  D(Mod-Gp-B) of triangulated categories. Proof. Let F : Db (mod-A) −→ Db (mod-B) be the derived equivalence with the quasi-inverse G. Let E be the set of all G(X), where X runs through isomorphism classes of finitely generated Gorenstein projective B-modules. We claim that E has the following properties: (i) HomKb (Gp-A) (E, E  [i]) = 0, for all E, E  ∈ E and i = 0. (ii) E generates Kb (Gp-A) as a triangulated category. (iii) E is equivalent to Gp-B. By Lemma 4.2.3, E is a subcategory of Kb (Gp-A). Also, Lemma 4.2.4 implies that condition (i) holds true. Moreover, two categories E and Gp-B are equivalent via G. Indeed, let X and Y be modules in Gp-B. Then there exist the following isomorphisms HomB (X, Y )

∼ = HomD(mod-A) (G(X), G(Y )) ∼ = HomK(mod-A) (G(X), G(Y )).

Note that the last isomorphism follows from Lemma 4.2.2. This means that the functor G : Gp-B −→ E is full and faithful. Also, clearly G is dense and hence is an equivalence. Therefore, we just need to prove that E generates Kb (Gp-A) as a triangulated category. First note that, since G is a derived equivalence, add-G(B) generates Kb (prj-A) as a triangulated category. Let Y be a Gorenstein projective A-module. Then there is a complete projective resolution ···

Q−n

∂ −n

···

Q−1

∂ −1

Q0

···

Qn

∂n

···

Y of Y with Qi ∈ prj-A. Set Y  := Ker∂ n . Since Y  ∈ Gp-A, by Lemma 4.2.3, F (Y  ) is isomorphic in Db (mod-B) to a complex 0 −→ TY 0 −→ · · · −→ TY n −→ 0 with TY 0 Gorenstein projective and TY i projective for every 1 ≤ i ≤ n. Consider the triangle F (Y  ) → TY 0  and apply the functor G on it to get the triangle

<1 F (Y



)→

(∗) G(<1 F (Y  )) → GF (Y  ) → G(TY 0 )  . Since <1 F (Y  ) is a bounded complex of finitely generated projective A-modules, G(<1 F (Y  )) is also a bounded complex of finitely generated projective B-modules. Moreover, by Lemma 4.2.3, G(TY 0 ) is isomorphic in Db (mod-A) to the following complex 0 −→ G−n −→ P −n+1 −→ · · · −→ P 0 −→ 0 with G−n ∈ Gp-A and P i ∈ prj-A for all −n + 1 ≤ i ≤ 0. Now, consider the image of triangle (∗) in Dbsg (A). We have isomorphisms G(TY 0 ) ∼ = G−n [n] ∼ = Y  = GF (Y  ) in Dbsg (A). Set X  := TY 0 and Z := G−n . Then

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there is a triangle Q → G(X  ) → Z[n]  in Kb (Gp-A), where Q is a bounded complex of projectives. This implies that Z[n] belongs to thick(E). Consider the triangle 

Y → Q → Y  [−n] , 

where Q is the truncated complex 0 −→ Q0 −→ · · · −→ Qn−1 −→ 0. ∼ Y , in Db (A). ∼ Y [n] in Db (A). Therefore, there is an isomorphism Z[n] ∼ Thus, Y  = = Y [n], and then Z = sg sg The full and faithful functor H : Gp-A −→ Dbsg (A) implies that Z ∼ = Y in Gp-A. So there exist projective modules P and Q such that Z ⊕ P ∼ = Y ⊕ Q in Gp-A. Now, since Z, P and Q all belong to thick(E), Y also belongs to thick(E). This finishes the proof of the claim. By Lemma 4.2.5, there is an equivalence ϕ¯ : Kb (Gp-A) −→ Kb (prj-(Gp-A)). Let T be the subcategory of b K (prj-(Gp-A)) that corresponds to E via ϕ. ¯ The properties of E, mentioned above, imply that T satisfies Conditions (P1) and (P2) of 3.1. Moreover, property (ii) of E implies that thick(T ) = Kb (prj-(Gp-A)). Now Theorem 3.15 comes to play to give an equivalence D(Mod-T )  D(Mod-(Gp-A)) of triangulated categories. Hence, by property (iii) of E, we get that D(Mod-(Gp-B))  D(Mod-(Gp-A)) as triangulated categories. The proof is hence complete. 2 4.2.7.

Let R be a ring. By [2, Theorem 3.3], for a ring R, we have an equivalence DbGp (Mod-R)  K−,Gpb (Add-(Gp-R)),

where K−,Gpb (Add-(Gp-R)) is the subcategory of K− (Add-(Gp-R)) consisting of all complexes X  in which there exists an integer n = n(X  ) such that Hi (Hom(G, X  )) = 0 for all i ≤ n and all G ∈ Gp-R. Corollary 4.2.8. Let A and B be right coherent rings that are derived equivalent. Then they are Gorenstein derived equivalent, i.e. DbGp (Mod-A)  DbGp (Mod-B). Proof. By Theorem 4.2.6, there is an equivalence ∼

Db (Mod(Gp-A)) −→ Db (Mod(Gp-B)) of triangulated categories. Also, the same argument as in the proof of Lemma 4.2.5 works to prove equivalences K−,Gpb (Add-(Gp-A))  K−,b (Prj-(Gp-A)) and K−,Gpb (Add-(Gp-B))  K−,b (Prj-(Gp-B)) of triangulated categories. Now, the result follows from 4.2.7 and the known triangulated equivalences Db (Mod(Gp-A))  K−,b (Prj-(Gp-A)) and Db (Mod(Gp-B))  K−,b (Prj-(Gp-B)). 2

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4.3. Infinitely generated n-tilting modules Infinitely generated tilting modules arise naturally to extend some aspects of the classical tilting theory to infinitely generated modules over arbitrary rings. The first instance of a generalization of Brenner and Butler’s theorem for infinitely generated tilting modules was given by Facchini [12] and [13]. Recently, infinitely generated tilting modules over arbitrary rings has received considerable attention towards knowing derived categories and equivalences of general rings, see [6,7,29]. Also, Chen and Xi [10] studied a relation between two concepts of good tilting modules and recollements of derived categories of rings. Let A be a ring. A (right) A-module M (possibly infinitely generated) is called n-tilting if the following conditions hold true: (i) pdA M ≤ n; (ii) for all i > 0 and every cardinal α, ExtiA (M, M (α) ) = 0; (iii) there exists an exact sequence 0 −→ A −→ M 0 −→ M 1 −→ · · · −→ M m −→ 0, with M i ∈ Add-M for 0 ≤ i ≤ m. An n-tilting module M is called a good n-tilting, if there exists an exact sequence 0 −→ A −→ M 0 −→ M 1 −→ · · · −→ M n −→ 0, where for every i, M i is isomorphic to a direct summand of finite direct sums of copies of M . Bazzoni et al. [7] proved the following important results on good n-titling modules. Proposition 4.3.1. ([7, Proposition 1.4]) Let MA be a good n-tilting module and B = EndA (M )op . Then (i) MB admits a projective resolution 0 −→ Qn −→ · · · −→ Q0 −→ MB −→ 0, in which for every 0 ≤ i ≤ n, Qi is finitely generated projective B-module; (ii) ExtiB (M, M ) = 0 for all i > 0; (iii) there exists a ring isomorphism EndB (M ) ∼ = A. We use this result to prove that every good n-tilting module MA provide an equivalence between the derived category D(Mod-A) and the relative derived category DM (Mod-EndA (M )op ). Theorem 4.3.2. Assume that M is a good n-tilting module and B = EndA (M )op . Then there exists an equivalence D(Mod-A)  DM (Mod-B), where DM (Mod-B) is the relative derived category with respect to {M }, see 3.11. Proof. In view of Proposition 4.3.1, there exists an exact sequence 0 −→ Qn −→ · · · −→ Q0 −→ M −→ 0,

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with Qi finitely generated projective B-modules, for every 0 ≤ i ≤ n. Let T  be the truncated complex 0 −→ Qn −→ · · · −→ Q0 −→ 0 of finitely generated B-modules. It can be easily checked, using Proposition 4.3.1, that T = {T  } satisfies Conditions (P1) and (P2) of 3.1. So there is a triangulated equivalence Φ : KKT (Mod-B) −→ KT (Mod-B). Now, observe that KKT (B-Mod) is equivalent to Kprj (Mod-End(T  )) via the functor HomKK(Mod-B) (T  , −). Also, Lemma 3.12 implies that KT (Mod-B)  DT (Mod-B). Therefore, we have an equivalence ∼

D(Mod-EndB (T  )) −→ DT (Mod-B) of triangulated categories. Since EndD(Mod-B) (T  ) ∼ = EndB (M ) and EndB (M ) ∼ = A by Proposition 4.3.1, there is the desired equivalence D(Mod-A)  DM (Mod-B) of triangulated categories. 2 Acknowledgements The authors would like to thank the referees for careful reading of the manuscript and many useful hints and comments that improved our exposition. This work was partially supported by Iran National Science Foundation (INSF) under contract No. 94028637. The first author thanks the Center of Excellence for Mathematics (University of Isfahan). Part of this work was done while the first author was visiting Max-Planck Institute for Mathematics-Bonn (MPIM). He would like to thank MPIM for support and the excellent atmosphere. References [1] J. Asadollahi, P. Bahiraei, R. Hafezi, R. Vahed, On relative derived module categories, Commun. Algebra 44 (2016) 5454–5477. [2] J. Asadollahi, R. Hafezi, R. Vahed, Gorenstein derived equivalences and their invariants, J. Pure Appl. Algebra 218 (2014) 888–903. [3] M. Auslander, Representation dimension of Artin algebras I, in: Selected Works of Maurice Auslander, Part 1, in: Queen Mary College Math. Notes, American Mathematical Society, 1971, pp. 505–574. [4] M. Auslander, Representation theory of Artin algebras. I, Commun. Algebra 1 (1974) 177–268. [5] M. Auslander, Applications of morphisms determined by modules, in: Representation Theory of Algebras, Proc. Conf., Temple Univ., Philadelphia, PA, 1976, in: Lect. Notes Pure Appl. Math., vol. 37, Dekker, New York, 1978, pp. 245–327. [6] S. Bazzoni, Equivalences induced by infinitely generated tilting modules, Proc. Am. Math. Soc. 138 (2010) 533–544. [7] S. Bazzoni, F. Mantese, A. Tonolo, Derived equivalence induced by infinitely generated n-tilting modules, Proc. Am. Math. Soc. 139 (2011) 4225–4234. [8] A. Beligiannis, Cohen–Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras, J. Algebra 288 (2005) 137–211. [9] D.J. Benson, H. Krause, Complexes of injective kG-modules, Algebra Number Theory 2 (2008) 1–30. [10] H. Chen, C.C. Xi, Good tilting modules and recollements of derived module categories, Proc. Lond. Math. Soc. (3) 104 (5) (2012) 959–996. [11] D. Dugger, B. Shipley, K-theory and derived equivalences, Duke Math. J. 124 (2004) 587–617. [12] A. Facchini, Divisible modules over integral domains, Ark. Mat. 26 (1) (1988) 67–85. [13] A. Facchini, A tilting module over commutative integral domains, Commun. Algebra 15 (11) (1987) 2235–2250. [14] N. Gao, P. Zhang, Gorenstein derived categories, J. Algebra 323 (2010) 2041–2057.

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[m3L; v1.236; Prn:30/05/2018; 12:57] P.24 (1-24) J. Asadollahi et al. / Journal of Pure and Applied Algebra ••• (••••) •••–•••

[15] W. Hu, S.Y. Pan, Stable functors of derived equivalences and Gorenstein projective modules, Math. Nachr. 290 (2017) 1512–1530. [16] W. Hu, C.C. Xi, Derived equivalences and stable equivalences of Morita type, I, Nagoya Math. J. 200 (2010) 107–152. [17] P. Jørgensen, Existence of Gorenstein projective resolutions and Tate cohomology, J. Eur. Math. Soc. 9 (1) (2007) 59–76. [18] B. Keller, Deriving DG categories, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994) 63–102. [19] H. Krause, Derived categories, resolutions, and Brown representability, in: Interactions Between Homotopy Theory and Algebra, in: Contemp. Math., vol. 436, Amer. Math. Soc., Providence, RI, 2007, pp. 101–139. [20] H. Krause, Localization theory for triangulated categories, in: Triangulated Categories, in: Lond. Math. Soc. Lect. Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 161–235. [21] H. Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005) 1128–1162. [22] J. Miyachi, Compact objects in stable module categories, Arch. Math. 89 (1) (2007) 47–51. [23] D. Murfet, Sh. Salarian, Totally acyclic complexes over Noetherian schemes, Adv. Math. 226 (2011) 1096–1133. [24] A. Neeman, The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Éc. Norm. Supér. (4) 25 (5) (1992) 547–566. [25] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Am. Math. Soc. 9 (1996) 205–236. [26] A. Neeman, Triangulated Categories, Ann. Math. Stud., vol. 148, Princeton University Press, Princeton, NJ, 2001. [27] A. Neeman, The homotopy category of flat modules, and Grothendieck duality, Invent. Math. 174 (2008) 255–308. [28] P. Nicolás, The bar derived category of a curved dg algebra, J. Pure Appl. Algebra 212 (2008) 2633–2659. [29] P. Nicolás, M. Saorín, Generalized tilting theory, Appl. Categor. Struct. 26 (2018) 309–368. [30] Sh. Pan, Derived equivalences for Cohen–Macaulay Auslander algebras, J. Pure Appl. Algebra 216 (2012) 355–363. [31] J. Rickard, Morita theory for derived category, J. Lond. Math. Soc. (2) 39 (2) (1989) 436–456. [32] S. Schwede, B. Shipley, Stable model categories are categories of modules, Topology 42 (2003) 103–153.