Relative Singularity Categories with Respect to Gorenstein Flat Modules

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1 Acta Mathematica Sinica, English Series Nov., 2017, Vol. 33, No. 11, pp Published online: August 29, Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2017 Relative Singularity Categories with Respect to Gorenstein Flat Modules Zhen ing DI Zhong Kui LIU Department of Mathematics, Northwest Normal University, Lanzhou , P. R. China dizhenxing @126.com liuzk@nwnu.edu.cn iao iang ZHANG Department of Mathematics, Southeast University, Nanjing , P. R. China z990303@seu.edu.cn Abstract Let R be a right coherent ring and D b (R-Mod) the bounded derived category of left R-modules. Denote by D b (R-Mod) [GF,C] the subcategory of D b (R-Mod) consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K b (F C) the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category D b (R-Mod) [GF,C] /K b (F C) is triangle-equivalent to the stable category GF C of the Frobenius category of all Gorenstein flat and cotorsion left R-modules. Keywords Triangle equivalence, Gorenstein flat dimension, cotorsion dimension, stable category, derived category, homotopy category MR(2010) Subject Classification 16E05, 16E10, 16E35 1 Introduction In 1987, Buchweitz [6] studied Verdier s quotient triangulated category D sg (R) := D b (Rmod)/K b (R-proj), where D b (R-mod) is the bounded derived category of finitely generated modules over a noetherian ring R and K b (R-proj) is the bounded homotopy category of finitely generated projective modules, under the name of stable derived category. He used this quotient category to study stable homological algebra and to define Tate cohomology for Gorenstein rings. In the representation theory of finite-dimensional algebras, this quotient category appeared in Rickard s work [24]. It is proved therein that this category is triangle-equivalent to the stable module category over a self-injective algebra. Later, this result was generalized to Gorenstein artin algebras via the (co)tilting theory by Happel [15]. Recently, Orlov [22] reconsidered this triangulated category and called D sg (R) thesingularity category of the ring R because this quotient category reflects certain homological singularity of the ring R. Besides, other quotient triangulated categories have also attracted increasing interest among scholars (we refer the reader to [7] for some basic knowledge about this topic). In particular, Received December 9, 2016, revised April 25, 2017, accepted May 23, 2017 Supported by National Natural Science Foundation of China (Grant Nos and ); the Postdoctoral Science Foundation of China (Grant No. 2106M602945B) and Northwest Normal University (Grant No. NWNU-LKQN-15-12)

2 1464 Di Z.. et al. Beligiannis studied the quotient triangulated category D b (R-Mod)/K b (R-Proj) for an arbitrary ring R, whered b (R-Mod) is the bounded derived category of R-modules and K b (R-Proj) is the bounded homotopy category of projective modules (see [2 4]). Just as the singularity category, this category reflects also the homological singularity of the ring R, and it treats modules which are not necessarily finitely generated. It seems like to call such a quotient triangulated category big singularity category of the ring R. Auslander [1] introduced Gorenstein dimension (abbr. G-dimension) for finitely generated modules over a commutative noetherian local ring. In particular, modules having G-dimension 0 can be regarded as a common generalization of finitely generated projective modules over commutative noetherian rings and maximal Cohen Macaulay modules over commutative Gorenstein local rings. Enochs and Jenda [10] called the modules having G-dimension 0 the Gorenstein projective and defined Gorenstein projective (whether finitely generated or not) modules over an arbitrary ring. Another important extension of the G-dimension is based on Gorenstein flat modules. These modules were introduced by Enochs et al. [12]. Further studies on Gorenstein flat modules can be found in [5, 8, 9, 11] et al.. It is well known that the subcategory GP of all Gorenstein projective modules together with all short exact sequences in GP form a Frobenius category with projective-injective objects all projective modules. This yields the stable category GP who kills all projective modules. By virtue of [16], we see that GP is also a triangulated category. In particular, it was shown in Beligiannis [2] that over a Gorenstein ring R there exists the triangle equivalence GP = D b (R-Mod)/K b (R-Proj), ( ) which is an extension of the case of finitely generated modules established by Happel [15]. Unless otherwise stated, in what follows R denotes a right coherent ring with identity. Let R-Mod denote the category of all left R-modules. By a module, we always mean a left R- module. The motivation of this paper is to establish a Gorenstein flat version of the above triangle equivalence. To achieve our goal, we have to overcome the following question firstly. Question 1.1 Let R be a right coherent ring. Which subcategory associated to the category GF of all Gorenstein flat modules can be taken to form a Frobenius category? It is proved in Section 4 that the subcategory GF C forms a Frobenius category with projective-injective objects all modules in F C(here, the symbol F (resp., C) stands for the subcategory of all flat (resp., cotorsion) modules, hence modules in F Care just flat cotorsion modules). This yields the stable category GF C who also possesses the structure of a triangulated category. Next, we turn our attention to the following question. Question 1.2 Let R be a right coherent ring. Who can play the role of? in the following triangle equivalence? GF C =? /K b (F C). In 2010, Iacob [19] defined a notion of Gorenstein flat dimension for complexes over a left GFclosed ring (see Definition 3.1 for details). Recently, Hu [18] investigated further the equivalent conditions on the finiteness of the Gorenstein flat dimension of a complex via flat-cotorsion and complete flat resolutions (see Definition 3.2). On the other hand, Ren and Liu [23] introduced a notion of cotorsion dimension of complexes via proper resolutions of complexes induced by

3 Relative Singularity Categories with Respect to Gorenstein Flat Modules 1465 the flat model structure on the category of complexes of modules proposed by Gillespie [14], and gave some equivalent characterizations on the finiteness of the cotorsion dimension of a complex. It is shown in our main result, Theorem 4.5, the subcategory D b (R-Mod) [GF,C] of D b (R-Mod) consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension is indeed the appropriate candidate for the role of?. As a consequence, if we further assume that the ring R is Gorenstein, then there exist the following triangle equivalences, which is in parallel to ( ), G(F C) = GF C = D b (R-Mod)/K b (F C), where G(F C) denotes the subcategory of all modules that can be viewed as cokernels of totally (F C)-acyclic complexes (see Lemma 4.8 and Corollary 4.9). The contents of this paper are summarized as follows. Section 2 contains some necessary notation and definitions for use throughout this paper. In Section 3, we further investigate the Gorenstein flat dimension and cotorsion dimension of complexes to make some preparations for the proof of Theorem 4.5. Finally, Section 4 is devoted mainly to giving our main results and their proofs. 2 Preliminaries In this section, we mainly recall some necessary notions and definitions, which will be used in the sequel; most of them are employing from [8, 14]. A complex of modules n+1 δ n+1 n δ n n 1 will be denoted simply by. The n-th cycle (resp., homology) module is defined as Kerδn (resp., Kerδn /Imδn+1) and denoted by Z n () (resp.,h n ()). Set C n () =Cokerδn+1. We always identify a module N with the complex 0 N 0,whereN is in degree zero and 0 elsewhere. A morphism f : Y of complexes is a family of morphisms f =(f n : n Y n ) n Z of modules satisfying δn Y f n = f n 1 δn for all n Z. The category of complexes of modules is denoted by C(R-Mod). We use subscripts, and to denote boundedness conditions. For example, C (R-Mod) is the subcategory of C(R-Mod) consisting of all bounded below complexes (in what follows, by the term subcategory we always mean a full additive subcategory closed under isomorphisms). A quasi-isomorphism f : Y is a morphism such that the induced map H n (f) :H n () H n (Y ) is an isomorphism for all n Z. The complexes and Y are equivalent [8, A.1.11, p. 164] and denoted by Y, if they can be linked by a sequence of quasi-isomorphisms with arrows in alternating directions. Given a complex, denoteby[1] the complex with ([1]) n = n 1 and differential δ [1] n = δ n 1. Thecomplex([1])[1] is denoted by [2], and inductively one can define [n] for all n Z. For integers n and m, thehard right-truncation, n,of at n and the hard left-truncation, m,of at m are given by δ n+2 δ n+1 n : n+2 n+1 n 0

4 1466 Di Z.. et al. and m : 0 m δ m m 1 δ m 1 m 2. The soft right-truncation, n,of at n and the soft left-truncation, m,of at m are given by and δn+2 δn+1 n : n+2 n+1 Z n () 0 m : 0 C m () δ δ m m 1 m 1 m 2. Let T be a subcategory of R-Mod. Denote by K b (T ) the bounded homotopy category with each complex constructed by modules in T. The derived category of R-Mod is denoted by D(R-Mod). The supremum and infimum of capture its homological position; they are defined as follows: sup =sup{s Z H s () 0} and inf =inf{i Z H i () 0}. Following the convention, set sup = and inf = if is exact. The subcategory D b (R-Mod) consists of all complexes with H i () =0for i 0. Let A be an abelian category and a subcategory of A. For an object M A,write M (resp., M 1 )ifext 1 A (M,) = 0 (resp., Ext1 A (M,) = 0) for each object. Dually, one can define M and M 1. Recall that a pair (, Y) of subcategories of A is called a cotorsion pair or cotorsion theory provided that = 1 Y and Y = 1. A cotorsion pair (, Y) issaidtobehereditary if Ext 1 A (, Y ) = 0 for all objects and objects Y Y. A morphism φ : M with is called an -preenvelope of M if for any morphism f : M with,thereis a morphism g : such that gφ = f. A monomorphism φ : M B with B is said to be a special -preenvelope of M if φ is an -preenvelope of M and Cokerφ 1. Dually, one has the definitions of an -precover and a special -precover. A cotorsion pair (, Y) is said to be complete provided that every object of A has a special Y-preenvelope and a special -precover. Perhaps the most useful complete hereditary cotorsion pair is the flat cotorsion pair (F, C). The book [11] is a standard reference for cotorsion pairs. Let P be a complex. Recall that P is called dg-projective if P i is a projective module for each i Z, and for each quasi-isomorphism f : Y of complexes, Hom R (P, f) :Hom R (P, ) Hom R (P, Y ) is also a quasi-isomorphism of abelian groups. Dually, one has the notion of dginjective complexes. Gillespie introduced in [14] the following definition associated to the flat cotorsion pair (F, C) inr-mod. Definition 2.1 ([14, Definition 3.3]) Let (F, C) be the flat cotorsion pair in R-Mod and a complex. (1) is called a flat complex if it is exact and Z i () F for each i Z. (2) is called a cotorsion complex if it is exact and Z i () C for each i Z. (3) is called a dg-flat complex if i Ffor each i Z, andhom R (, C) is exact whenever C is a cotorsion complex.

5 Relative Singularity Categories with Respect to Gorenstein Flat Modules 1467 (4) is called a dg-cotorsion complex if i C for each i Z, andhom R (F, ) is exact whenever F is a flat complex. The class of all flat (resp., dg-flat) complexes is denoted by F (resp., dg F). Similarly, the class of all cotorsion (resp., dg-cotorsion) complexes is denoted by C (resp., dg C). Clearly, any dg-projective (resp., dg-injective) complex is dg-flat (resp., dg-cotorsion) by the definitions involved. By virtue of [14, Lemma 3.4], any complex A C (R) (resp.,a C (R)) with all entries in F (resp., C) is a dg-flat (resp., dg-cotorsion) complex. The following result is cited from [18]. It will be used in the proofs of Lemmas 3.5 and 3.6. Lemma 2.2 ([18, Lemma 3.11]) Let be a complex. (1) If inf {i Z i 0} k with k an integer, then there exists a special dg-flat precover π : F of with F i =0for all i<k. (2) If sup {i Z i 0} k with k an integer, then there exists a special dg-cotorsion preenvelope q : C of with C i =0for all i>k. Let (F, C) be the flat cotorsion pair in R-Mod. Gillespie showed that both (dg F, C) and ( F,dg C) are cotorsion pairs in C(R-Mod), and they are called induced cotorsion pairs (see [14, Corollary 3.8 and Definition 3.7]). By virtue of [14, Corollary 3.12], we see that dg F E= F and dg C E= C, wheree denotes the class of all exact complexes. Lemma 2.3 ([14, Corollaries 3.13]) Let (F, C) be the flat cotorsion pair in R-Mod. Then the induced cotorsion pairs (dg F, C) and ( F, dg C) are both complete. Foracomplex, recall from [18, Definition 3.1] that a flat-cotorsion resolution of is a diagram F π C q of morphisms of complexes with q a special dg-cotorsion preenvelope and π a special dg-flat precover (note that in this case the complex F is in dg F dg C). In view of Lemma 2.3, we conclude that every complex admits flat-cotorsion resolutions. 3 Gorenstein Flat Dimension and Cotorsion Dimension of Complexes To prove our main result, Theorem 4.5, appearing in the next section we investigate further the Gorenstein flat dimension and the cotorsion dimension of a complex. In particular, Lemma 3.6 will play a key role in the proof of Theorem 4.5. Recall that a module M is called Gorenstein flat [11] if there exists an exact sequence F 1 F 0 F 1 F 2 of flat modules such that M = Ker(F 1 F 2 )andi R leaves the sequence exact whenever I is an injective right R-module. The subcategory of all Gorenstein flat modules is denoted by GF. In 2009, Iacob introduced in [19] the following notion of Gorenstein flat dimension for complexes over left GF-closed rings which are the rings for which GF is closed under extensions. According to [5], the class of left GF-closed rings includes strictly the one of left coherent rings. Definition 3.1 ([19]) Let M be a complex and n an integer. The Gorenstein flat dimension Gfd R (M) of M is defined as follows.

6 1468 Di Z.. et al. Gfd R (M) n if there is a quasi-isomorphism F M with F dg-flat such that sup F n and C j (F ) is Gorenstein flat for any integer j n. If Gfd R (M) n but Gfd R (M) n 1 does not hold, then Gfd R (M) =n. If Gfd R (M) m for any integer m, thengfd R (M) =. If Gfd R (M) m does not hold for any integer m, thengfd R (M) =. Fact 1 For a complex M andanintegern, by [19, Theorem 1], we see that Gfd R (M) n if and only if sup M n and C n (P ) is Gorenstein flat for any (some) dg-projective complex P with P M. In particular, this fact implies that the subcategory of all complexes with finite Gorenstein flat dimension is closed under equivalences of complexes. To establish a theory of Tate Vogel cohomologies of complexes based on flat objects, Hu [18] introduced recently the following interesting resolutions for complexes over arbitrary rings. Definition 3.2 ([18]) Let M be a complex. A complete flat resolution of M is a diagram T τ F π C q M of morphisms of complexes satisfying : (1) F π C q M is a flat-cotorsion resolution of M. (2) T is an exact complex with each entry in F Cand Z i (T ) GF for each i Z. (3) τ : T F is a morphism such that τ i =id Ti for all i 0. It was shown by Hu [18] that flat-cotorsion and complete flat resolutions are closely related to the finiteness of Gorenstein flat dimension of complexes. Fact 2 For a complex M andanintegern, according to [18, Proposition 5.4], we see that Gfd R (M) n if and only if sup M n and C n (F ) is Gorenstein flat for any flat-cotorsion resolution F C M of M if and only if for each flat-cotorsion resolution F C M of M, there exists a complete flat resolution T τ F C M of M such that τ i =id Ti for all i n. Recently, Ren and Liu [23] introduced the following notion of cotorsion dimension for complexes over arbitrary rings. Definition 3.3 ([23]) Let M be a complex. The cotorsion dimension of M, denoted by dg Cid(M), is defined as dg C-id(M) =inf{sup{i C i =0} M C with C dg C}. If dg C-id(M) n for all n Z, wewritedg C-id(M) =. If dg C-id(M) n for no n Z, wewritedg C-id(M) =. It is clear that dg C-id(M) = if and only if M is an exact complex. Fact 3 For a complex M and an integer n, according to [23, Theorem 3.3], we see that dg Cid(M) n if and only if inf M n and Z n (I) Cfor any (some) dg-injective complex I with I M if and only if inf M n and Z n (C) Cfor any (some) special dg-cotorsion preenvelope M C of M. In particular, one can easily deduce by these facts that the subcategory of all complexes with finite cotorsion dimension is closed under equivalences of complexes. In view of Fact 3, one can obtain the following result by a similar argument to the dual version of [26, Theorem 3.9].

7 Relative Singularity Categories with Respect to Gorenstein Flat Modules 1469 Lemma 3.4 Let 0 M M M 0 be a short exact sequence of complexes. If any two complexes of M, M and M have finite cotorsion dimension, then so does the third. Lemma 3.5 Let C be a complex such that C dg C C (R-Mod) and n an integer. If Gfd R (C) n, then there exists a complete flat resolution T τ F C id C C of C such that Z i (T ) GF C for each i Z, τ i =id Ti for all i n, andf C (R-Mod). Proof Note that C C (R-Mod) by assumption. It follows from Lemma 2.2 (1) that C admits a special dg-flat precover F C with F C (R-Mod), that is, there exists a short exact sequence 0 H F C 0 ( ) in C(R-Mod) with F C (R-Mod) and H a cotorsion complex. Since C belongs to dg C as well, it is trivial that C id C C is a special dg-cotorsion preenvelope of C. Therefore, we have a flat-cotorsion resolution F C id C C of C. On the other hand, since Gfd R (C) n by assumption, we deduce by [18, Proposition 5.4] (see also Fact 2) that there exists a complete flat resolution T τ F C id C C of C such that τ i =id Fi for all i n. To complete the proof, it remains to show that each Z i (T )isalsoinc. To this end, let t = max{m, n}, wherem =sup{i Z C i 0}. Applying the functor Hom R (G, ) to( ) withg any modules in F, wehavehom(g, F ) Hom(G, C). This yields that for each i t, theexact sequence 0 Z i+1 (F ) F i+1 Z i (F ) 0 in R-Mod is Hom R (G, )-exact. Hence, Ext 1 R (G, Z i+1(f )) vanishes in the following exact sequence Hom R (G, F i+1 ) Hom R (G, Z i (F )) Ext 1 R(G, Z i+1 (F )) Ext 1 R(G, F i+1 )=0. It follows that Z i+1 (F ) Cfor each i t. NotethatT is of the form F t+1 F t T t 1 T t 2 with each entry belongs to C and C is closed under cokernels of monomorphisms. It follows that Z t (T )=Ker(F t T t 1 ) Cand Z j (T ) Cfor all j t 1. According to what we showed above, we see that each Z i (T ) belongs to C, as desired. This completes the proof. To prove that the triangle functor appearing in Theorem 4.5 is essentially surjective, we need the following result. Lemma 3.6 Let M be a complex and n, t integers. Assume that Gfd R (M) =n and dg C-id(M) = t. Then there exists a complex C satisfying : (1) C dg C C (R-Mod); (2) C M;

8 1470 Di Z.. et al. (3) C admits a complete flat resolution T τ F C id C C such that Z i (T ) GF C for each i Z, τ i =id Fi for all i n and F C (R-Mod). Proof Note that Gfd R (M) =n by assumption. It follows from [19, Theorem 1] (see also Fact 1) that sup M n. This implies M M n. AccordingtoLemma2.2(2),M n admits a special dg-cotorsion preenvelope M n C with C i =0foralli>n, i.e., C C (R-Mod). Meanwhile, we have M n C. On the other hand, since dg C-id(M) =t by assumption, we deduce from [23, Theorem 3.3] (see also Fact 3) that dg C-id(M n ) t. In view of [23, Theorem 3.3] (see also Fact 3) again, we know that inf C =infm n t and Z t (C ) C. These facts imply that C C t and C t is a bounded complex with all entry belong to C. Thus, C t dg C C (R-Mod). According to what we showed above we see also that M C t. Hence, in view of [19, Theorem 1] (see also Fact 1), we have Gfd R (C t) n. Next, we wish to show that C t admits a complete flat resolution as described in (3). If we can achieve the goal, it is clear that C t is the appropriate candidate for C requested in the lemma. Indeed, Lemma 3.5 guarantees this fact. This completes the proof. 4 Main Results In this section we establish the desired triangle equivalences on the Gorenstein flat modules as described in Introduction. Let A be an abelian category and, W subcategories of A. Following Sather-Wagstaff et al. [25], the subcategory W is said to be a cogenerator for if W, and for each object, there exists a short exact sequence 0 W 0 in A such that W W and. The subcategory W is called an injective cogenerator for if W is a cogenerator for and W. Lemma 4.1 ([9, Lemma 4.2]) The subcategory F Cis an injective cogenerator for GF. Let A be an essentially small exact category, that is, an additive full subcategory of an abelian category which is closed under extensions. If A has enough projectives and injectives, and the projectives coincide with the injectives, then A is called a Frobenius category. Denote the class of all projective-injective objects of A by I. Then the stable category A/I is a triangulated category (see [16] for details). Recall that the distinguished triangles in A/I are induced by the pushout as follows. Let 0 i I() T () 0 u 0 Y C(u) T () 0 be a commutative diagram in A with exact rows in which I() is an injective object, C(u) is the pushout of (i, u) andt () is the first cosyzygy of. Then the sequence u Y C(u) T () is a distinguished triangle in A/I with T the shift functor in A/I.

9 Relative Singularity Categories with Respect to Gorenstein Flat Modules 1471 Proposition 4.2 The subcategory GF C forms a Frobenius category whose projective-injective objects are precisely all modules in F C. Proof According to [17, Propositiom 3.12], we see that GF is closed under extensions. Hence so is GF C. ThenGF C becomes an exact category whose conflations are just short exact sequences with all terms in GF C (see Example 4.1 in [20]). We will show firstly that modules in F Care projective and injective in GF C. Tothis end, according to Lemma 4.1, we see that (GF C) (F C). This implies that modules in F Care injective. On the other hand, it is trivial that (F C) (GF C). Hence modules in F Care projective. Let M be any module in GF C. By Lemma 4.1 again, there exists an exact sequence 0 M K M 0inR-Mod with K F Cand M GF C. This shows that the exact category GF C has enough injective objects. On the other hand, since M C, it follows from [11, Lemma ] that there exists an exact sequence 0 M K M 0 in R-Mod with K F Cand M C.NotethatMbelongs to GF as well. We conclude that M is also in GF because GF is closed under kernels of epimorphisms. Therefore, M GF C. This implies that the exact category GF C has enough projective objects. From the argument above, it is direct to conclude that in the exact category GF C the class of projective objects coincides with the class of injective objects, and projective-injective objects are just modules in F C. We need the following two easy facts; it will be applied in the proof of Theorem 4.5. Lemma 4.3 Let M be a module in GF C. (1) If F is a complex in K b (F C) such that F i =0for i 0, thenhom D(R-Mod) (M,F) =0. (2) If F is a complex in K b (F C) such that F i =0for i 0, thenhom D(R-Mod) (F, M) =0. Proof We only prove the case (1); the second can be obtained by a similar argument. To this end, apply the functor Hom D(R-Mod) (M, ) to the distinguished triangle F 1 [1] F F 2 F 1 [2] in K(R-Mod). Note that for any integer i 1, we have Hom D(R-Mod) (M,F 1 [i]) = Ext i R(M,F 1 )=0 by Lemma 4.1. Hence, the assertion can be get by using induction on the width of F. Proposition 4.4 All homology bounded complexes with both finite Gorenstein flat dimension and cotorsion dimension form a triangulated full subcategory of D b (R-Mod). In what follows, it is denoted by D b (R-Mod) [GF,C]. Proof Let M and M be two homology bounded complexes such that M = M in D b (R-Mod). Assume that M has both finite Gorenstein flat dimension and cotorsion dimension. Then in view of Facts 1 and 3, we see that M has the same properties as M. This implies that D b (R-Mod) [GF,C] is closed under isomorphisms in D b (R-Mod). Moreover, it is clear that D b (R-Mod) [GF,C] is closed under shifts. Hence it remains to show that D b (R-Mod) [GF,C] is closed under cones. To this end, let Y Z [1]

10 1472 Di Z.. et al. be a distinguished triangle in D b (R-Mod). We may assume that it is induced by a short exact sequence 0 Y Z 0 in C(R-Mod). Now the assertion follows by [19, Proposition 4] and Lemma 3.4. Let T be a triangulated category and K a triangulated subcategory of T closed under summands, that is, a thick subcategory. Then one can form the triangulated quotient T /K (see [13]). It is also a triangulated category. According to Proposition 4.4, we know that D b (R-Mod) [GF,C] is a triangulated category. Moreover, it is clear that K b (F C) is a triangulated subcategory of D b (R-Mod) [GF,C],which is closed under direct summands. Thus the triangulated quotient D b (R-Mod) [GF,C] /K b (F C) is also a triangulated category. Notice that any module in GF C as a complex has both finite Gorenstein flat dimension and cotorsion dimension, so there exists an embedding: GF C D b (R-Mod) [GF,C]. Let F be the composition: GF C D b (R-Mod) [GF,C] D b (R-Mod) [GF,C] /K b (F C), where the latter one is the natural quotient functor. It is clear that F sends modules in F C to0ind b (R-Mod) [GF,C] /K b (F C), so it factors through the stable category GF C (see Proposition 4.2). Consequently, there exists a functor F : GF C D b (R-Mod) [GF,C] /K b (F C) such that F = Fπ,whereπ : GF C GF C is the natural quotient functor. Now, we are ready to give our main result of the paper. Theorem 4.5 The functor F : GF C D b (R-Mod) [GF,C] /K b (F C) is a triangle equivalence. Proof We want to show that F is a triangle functor, and it is essentially surjective (or dense), full and faithful. (1) F is a triangle functor. Let u Y Z T () be a distinguished triangle in GF C. Then it comes from a commutative diagram 0 I() T () 0 u 0 Y Z T () 0 in GF C with exact rows. This yields a commutative diagram of distinguished triangles I() T () [1] u u[1] Y Z T () Y [1]

11 Relative Singularity Categories with Respect to Gorenstein Flat Modules 1473 in D b (R-Mod) [GF,C]. Send it now to a commutative diagram in D b (R-Mod) [GF,C] /K b (F C). Then we have T () = [1] since I() is0ind b (R-Mod) [GF,C] /K b (F C). Thus, u Y Z [1] is a distinguished triangle in D b (R-Mod) [GF,C] /K b (F C). It follows that F is a triangle functor. (2) F is essentially surjective (or dense). Let M be any complex in D b (R-Mod) [GF,C] /K b (F C). Assume that Gfd R (M) =n and dg C-id(M) =t for some integers n and t. Then in view of Lemma 3.6, there exists a complex C dg C C (R-Mod) satisfying C = M in D b (R-Mod) and C admits a complete flat resolution T τ F C id C C such that Z i (T ) GF Cfor each i Z, τ i =id Fi for all i n and F C (R-Mod). Note that F is of the form We have a distinguished triangle F n+1 F n F t+1 F t 0. F n 1 F F n F n 1 [1] in K(R-Mod). Send it now to a distinguished triangle in D b (R-Mod) [GF,C] /K b (F C). Therefore, F = F n in D b (R-Mod) [GF,C] /K b (F C). Moreover, it is easy to see that F n = Cn (F )= Z n 1 (T )ind b (R-Mod). This implies F = Z n 1 (T )ind b (R-Mod) [GF,C] /K b (F C). Note that Z n 1 (T ) belongs to GF C. It follows that F is essentially surjective. (3) F is full. Since we have F = Fπ, it suffices to show that F is full. Let f Z g Y be a morphism in D b (R-Mod) [GF,C] /K b (F C)with, Y GF Cand f lies in the compatible saturated multiplicative system corresponding to K b (F C). Complete f to a distinguished triangle [ 1] ω Q Z f with Q K b (F C). Consider the distinguished triangle Q 1 h Q ϕ Q 0 Q 1 [1] in K b (F C). In view of Lemma 4.3 (1), we see that Hom D(R-Mod) ([ 1],Q 0 )=0. Thisyields ϕω = 0. Hence ω factors through h. Consider now the following commutative of distinguished triangles [ 1] Q 1 Z s [ 1] h ω Q Z f where s, l, f are all in the compatible saturated multiplicative system corresponding to K b (F C). Since Hom D(R-Mod) (Q 1,Y) = 0 by Lemma 4.3 (2), there exists some k : Y such that gl = ks = kfl. Sowehavek = gf 1.Thus,F is full, as desired. l

12 1474 Di Z.. et al. (4) F is faithful. Suppose that there exists a morphism f : Y in GF C such that F (f) =0. Wewant to show f = 0. To this end, complete f to a distinguished triangle f Y g Z [1] in GF C. Since F (f) =0,F (g) is a section. According to (3), we know that F is full. So there exists some morphism α : Z Y such that 1 F (Y ) = F (αg). Let β = αg and complete β to a distinguished triangle Y β Y C(β) Y [1] in GF C. WehaveF (C(β)) K b (F C). According to [11, Corollary ], we know that any Gorenstein flat module with finite flat dimension is flat. It follows that C(β) F C. Hence β is an isomorphism in GF C. Thisimpliesthatg is a section, and hence f =0,as desired. This completes the proof. RecallthataringissaidtobeGorenstein if it is left and right noetherian and has finite self injective dimension on both sides (see [11, Definition 9.1.1]). By virtue of [19, Theorem 2], we see that over a Gorenstein ring every homology bounded complex has finite Gorenstein flat dimension. On the other hand, note that over a Gorenstein ring each flat module has finite projective dimension. It follows from [21, Corollary ] that the left global cotorsion dimension of the ring, which is defined as the supremum of cotorsion dimension of modules, is finite. Hence, according to [23, Theorem 4.2], we know that every homology bounded complex has finite cotorsion dimension. These facts enable us to get the next result. Lemma 4.6 Let R be a Gorenstein ring. Then we have D b (R-Mod) [GF,C] = D b (R-Mod). As a consequence of the above lemma, we obtain the following triangle equivalence by Theorem 4.5. Corollary 4.7 Let R be a Gorenstein ring. Then there exists a triangle equivalence GF C = D b (R-Mod)/K b (F C). Let be a subcategory of R-Mod. Recall from [25, Definition 4.1] that an exact complex of modules in is called totally -acyclic if it is Hom R (, )-exact and Hom R (, )-exact. Denote by G( ) the subcategory of R-Mod whose modules are of the form M = Z 1 () for some totally -acyclic complex. Lemma 4.8 Let R be a Gorenstein ring. Then we have G(F C)=GF C. Proof In view of Proposition 4.2, it is easy to see that GF C G(F C). Hence, it remains to show that the converse containment holds. Let M be a module in G(F C). Then there exists a totally (F C)-acyclic complex F such that M = Z 1 (F ). For any injective right R-module I, by [11, Lemma ], we see that Hom Z (I,Q/Z) belongs to F C. Hence, the complex Hom Z (I R F, Q/Z) = Hom R (F, Hom Z (I,Q/Z)) is exact. This implies that the complex I R F is exact. Consequently, we have M GF.

13 Relative Singularity Categories with Respect to Gorenstein Flat Modules 1475 On the other hand, we wish to show that M belongs to C as well. To this end, let G be any module in F. NotethatR is Gorenstein by assumption. The projective dimension of G is finite. Therefore, it is easy to check that the complex Hom R (G, F ) is exact. Consequently, the short exact sequence 0 M F 1 Z 2 (F ) 0 in R-Mod is Hom R (G, )-exact. This yields that Ext 1 R (G, M) vanishes in the following exact sequence Hom R (G, F 1 ) Hom R (G, Z 2 (F )) Ext 1 R(G, M) Ext 1 R(G, F 1 )=0. It follows that M C, as desired. This completes the proof. We end the paper with the following result, which is a consequence of Lemma 4.8 and Corollary 4.7. One can compare it with [2, Theorem 6.9]. Corollary 4.9 Let R be a Gorenstein ring. Then there exists a triangle equivalence G(F C) = D b (R-Mod)/K b (F C). Acknowledgements We thank the referees for careful reading of the paper and for many valuable comments and suggestions. References [1] Auslander, M., Bridger, M.: Stable Module Theory, Memoirs Amer. Math. Soc., 94, Providence, Amer. Math. Soc., 1969 [2] Beligiannis, A.: The homological theory of contravariantly finite subcategories: Auslander Buchweitz contexts, Gorenstein categories and (co)stabilization. Comm. Algebra, 28, (2000) [3] Beligiannis, A.: Homotopy theory of modules and Gorenstein rings. Math. Scand., 89, 5 45 (2001) [4] Beligiannis, A.: Cohen Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras. J. Algebra, 288, (2005) [5] Bennis, D.: Rings over which the class of Gorenstein flat modules is closed under extensions. Comm. Algebra, 37, (2009) [6] Buchweitz, R. O.: Maximal cohen-macaulay modules and Tate cohomology over Gorenstein rings. Unpublished manuscript, 155pp (1987). Available at [7] Chen,. W.: Relative singularity categories and Gorenstein projective modules. Math. Nachr., 284, (2011) [8] Christensen, L. W.: Gorenstein Dimensions, Lecture Notes in Math., Springer-Verlag, Berlin, 2000 [9] Di, Z.., Wei, J. Q., Zhang,.., et al.: On balance for relative homology. Comm. Algebra, 44, (2016) [10] Enochs, E. E., Jenda, O. M. G.: Gorenstein injective and projective modules. Math. Z., 220, (1995) [11] Enochs, E. E., Jenda, O. M. G.: Relative Homological Algebra, Vol. 30. de Gruyter, Berlin: de Gruyter Exp. Math., 2000 [12] Enochs, E. E., Jenda, O. M. G., Torrecillas, B.: Gorenstein flat modules. Journal Nanjing Univ., 10, 1 9 (1993) [13] Gelfand, S. I., Manin, Y. I.: Methods of Homological Algebra, Second edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003 [14] Gillespie, J.: The flat model structure on Ch(R). Trans. Amer. Math. Soc., 356, (2004) [15] Happel, D.: On Gorenstein Algebras, Progress in Mathematics Vol. 95, Birkh auser Verlag, Basel, 1991, [16] Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988

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