Gorenstein Algebras and Recollements

Gorenstein Algebras and Recollements Xin Ma 1, Tiwei Zhao 2, Zhaoyong Huang 3, 1 ollege of Science, Henan University of Engineering, Zhengzhou 451191, Henan Province, P.R. hina; 2 School of Mathematical Sciences, ufu Normal University, ufu 273165, Shandong Province, P. R. hina; 3 Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P.R. hina Abstract Let A, A and A be artin algebras. We prove that if there is a recollement of the bounded Gorenstein derived category D b G(P(Mod A))(Mod A) relative to the bounded Gorenstein derived categories D b G(P(Mod A ))(Mod A ) and D b G(P(Mod A ))(Mod A ), then A is Gorenstein if and only if so are A and A. In addition, we prove that a virtually Gorenstein algebra A is Gorenstein if and only if the bounded homotopy category of (finitely generated) projective left A-modules and that of (finitely generated) injective left A-modules coincide. 1 Introduction Recollements play an important role in algebraic geometry and representation theory, see [1, 9, 25], which were introduced by Beĭlinson, Bernstein and Deligne in [9]. It is known that there are many interesting homological properties or invariants in the framework of recollements. For instance, for a recollement of bounded derived categories over finite dimensional algebras, Wiedemann proved in [32] that the finiteness of the global dimension is invariant in the recollement, and Happel proved in [21] that the finiteness of the finitistic dimension is also invariant in the recollement, and so on. For an artin algebra A, we use Mod A and mod A to denote the category of left A-modules and the category of finitely generated left A-modules respectively. Let X = Mod A or X = mod A. We use G(P(X )) to denote the subcategory of X consisting of Gorenstein projective modules, and use DG(P(X b ))(X ) to denote the bounded Gorenstein derived categories of X ([19]). Recently, the Gorensteinness of algebras in recollements was studied by several authors. Let A, A and A be finite dimensional algebras and the bounded derived category D b (mod A) admit a recollement D b (mod A i ) i D b (mod A) j! D b (mod A ). Pan proved that if A is Gorenstein, then A and A are also Gorenstein ([28, Theorem 3.1]). Later, hen and König proved that A is Gorenstein if and only if A and A are also Gorenstein plus an extra condition ([15, Proposition 3.4]). in and Han proved that A is Gorenstein if and only if so are A and A provided the recollement is what they call a 4-recollement, where they used the unbounded derived category of Mod ([29, Theorem III]). On the other hand, Asadollahi, Bahiraei, Hafezi and Vahed studied 2010 Mathematics Subject lassification: 18G25, 18E30. Keywords: Recollements, Gorenstein algebras, Virtually Gorenstein algebras, Perfect objects, Relative derived categories. orresponding author Email addresses: maxin0719@126.com (X. Ma), tiweizhao@hotmail.com (T. W. Zhao), huangzy@nju.edu.cn (Z. Y. Huang) 1

2 X. Ma, T.W. Zhao, Z.Y. Huang the Gorensteinness of algebras in recollements of relative derived categories and they showed that the Gorensteinness of certain artin algebras is an invariant of recollements; that is, for artin algebras A, A and A with G(P(mod A)), G(P(mod A )) and G(P(mod A )) contravariantly finite in mod A, mod A and mod A respectively, if DG(P(mod b A)) (mod A) admits a recollement i DG(P(mod b A )) (mod A ) i DG(P(mod b A)) (mod A) j! DG(P(mod b A )) (mod A ), then A is Gorenstein if and only if so are A and A ([3, Theorem 4.7]). Independently, using a different proof from that in [3], Gao showed in [18, Theorem 3.5] that the Gorensteinness of virtually Gorenstein artin algebras is an invariant of recollements; that is, for virtually Gorenstein artin algebras A, A and A, if DG(P(mod b A)) (mod A) admits a recollement as above, then A is Gorenstein if and only if so are A and A. The following is one of the main results in this paper, which is a big module analogue of the Asadollahi, Bahiraei, Hafezi and Vahed s result mentioned above as well as a Gorenstein analogue of a classical result of Wiedemann ([32, Lemma 2.1]). Theorem 1.1. (orollary 3.8) For artin algebras A, A and A, if DG(P(Mod b A)) (Mod A) admits a recollement i DG(P(Mod b A )) (Mod A ) i DG(P(Mod b A)) (Mod A) then A is Gorenstein if and only if so are A and A. j! DG(P(Mod b A )) (Mod A ), In fact, we prove this result in some more general setting (Theorem 3.7). Let A be an artin algebra and X = Mod A or X = mod A. For a subcategory of X, we use K b () to denote the bounded homotopy category of. Happel proved in [20, Lemma 1.5] that a finite-dimensional algebra A is Gorenstein if and only if K b (I(mod A)) = K b (P(mod A)), where I(mod A) and P(mod A) are the subcategories of mod A consisting of injective modules and projective modules respectively. On the other hand, as an important generalization of Gorenstein algebras, Beligiannis and Reiten introduced in [11] virtually Gorenstein algebras. Note that any Gorenstein algebra is virtually Gorenstein and the converse is not true in general ([11, 12]). We get some equivalent characterizations for virtually Gorenstein artin algebras being Gorenstein, which can be regarded as a Gorenstein analogue of the above result of Happel. Theorem 1.2. (Theorem 4.6) For a virtually Gorenstein artin algebra A, the following statements are equivalent. (1) A is Gorenstein. (2) K b (G(P(Mod A))) = K b (G(I(Mod A))) in DG(P(Mod b A)) (Mod A). (3) K b (G(P(mod A))) = K b (G(I(mod A))) in DG(P(mod b A)) (mod A). 2 Preliminaries Throughout this paper, A is an abelian category and all subcategories of A are additive, full and closed under isomorphisms. We use P(A) and I(A) to denote the subcategories of A consisting of projective and injective objects, respectively. We fix a subcategory of A. A complex X in A is called -acyclic (resp. -coacyclic) if H i Hom A (, X) = 0 (resp. H i Hom A (X, ) = 0) for any and i Z (the set of integers).

Gorenstein Algebras and Recollements 3 Definition 2.1. ([6]) Let D be subcategories of A. The morphism f : D in A with and D D is called a right -approximation of D if the complex f D 0 is -acyclic. If every object in D admits a right -approximation, then is called contravariantly finite in D. Dually, the notions of left -approximations and covariantly finite subcategories are defined. Let K-ac (A) (resp. K -coac (A)) denote the subcategory of the homotopy category K (A) consisting of -acyclic (resp. -coacyclic) complexes, where {blank,, +, b}. A chain map f : X Y is called a -quasi-isomorphism (resp. -coquasi-isomorphism) if Hom A (, f) (resp. Hom A (f, )) is a quasiisomorphism for any. Let be a subcategory of A. Then the relative derived category, denoted by D (A) (resp. co-d (A)), is the Verdier quotient of the homotopy category K (A) with respect to the thick subcategory K-ac (A) (resp. K -coac (A)) ([14]). Set K,b () := {X K () there exists N Z such that H i Hom A (, X) = 0 for any and i N}. K +,b () := {X K + () there exists N Z such that H i Hom A (X, ) = 0 for any and i N}. Lemma 2.2. ([19, Theorem 3.6] and [4, Theorem 3.3]) (1) If is contravariantly finite in A, then D b (A) = K,b (). (2) If is covariantly finite in A, then co-d b (A) = K +,b (). Definition 2.3. ([31]) The Gorenstein subcategory G() of A is defined as G() = {M A there exists an exact sequence: 1 0 0 1 in A with all i, i in, which is both -acyclic and -coacyclic, such that M = Im( 0 0 )}. If = P(A) (resp. I(A)), then G(P(A)) (resp. G(I(A))) is exactly the subcategory of A consisting of Gorenstein projective (resp. Gorenstein injective) objects ([16]). Let A be an artin algebra. Recall from [10, 11] that A is called virtually Gorenstein if where and G(P(Mod A)) = G(I(Mod A)), G(P(Mod A)) = {M Mod A Ext 1 A (G, M) = 0 for any G G(P(Mod A))} G(I(Mod A)) = {M Mod A Ext 1 A (M, H) = 0 for any H G(I(Mod A))}. Definition 2.4. ([5, 7]) Let M A and n 0. The -dimension -dim M of M is said to be at most n if there exists an exact sequence 0 n n 1 0 M 0 (2.1) in A with all i objects in. Moreover, the sequence (2.1) is called a proper -resolution of M if it is -acyclic. Dually, The -codimension -codim M of M is said to be at most n if there exists an exact sequence 0 M 0 n 1 n 0 (2.2) in A with all i objects in. Moreover, the sequence (2.2) is called a coproper -coresolution of M if it is -coacyclic.

4 X. Ma, T.W. Zhao, Z.Y. Huang Definition 2.5. ([9]) Let T, T and T be triangulated categories. A recollement of T relative to T and T is a diagram T i j! i T j T of triangle functors such that (1) (i, i ), (i, ), (j!, ) and (, ) are adjoint pairs. (2) i = 0. (3) i, j! and are fully faithful. (4) For any object X in T, there exist triangles i X X X (i X)[1] and j! X X i i X (j! X)[1], where the maps are given by adjunctions. 3 Relative Singularity ategories and Recollements Definition 3.1. ([27]) Let D be a triangulated category. An object P D is called perfect if for any object Y D, Hom D (P, Y [i]) = 0 except for finitely many i Z. Dually, the notion of coperfect objects is defined. We use D perf (resp. D coperf ) to denote the triangulated subcategory consisting of perfect (resp. coperfect) objects, which is called the perfect (resp. coperfect) subcategory of D ([27]). Obviously, D perf and D coperf are thick subcategories of D, and both of them are invariants of triangle equivalence. Let be a contravariantly finite subcategory of A. Li and Huang introduced in [26] the relative singularity category by the following Verdier quotient D b -sg(a) = D b (A)/K b (). On the other hand, Orlov [27] defined it in a different way. Definition 3.2. ([27, Definition 1.7]) Let D be a triangulated category, the singularity category of D is defined by the Verdier quotient D/D perf. In particular, let A be an abelian category and a subcategory of A. The relative singularity category is defined by the following Verdier quotient D b -sg(a) = D b (A)/D b (A) perf. In general, we have K b () D b (A) perf. In the following, we will give some sufficient condition such that they are identical. In this case, the two definitions of the relative singularity categories mentioned above coincide. Rickard proved in [30, Proposition 6.2] that D b (Mod A) perf = K b (P(Mod A)) for any ring A. We generalize this result and [13, Lemma 1.2.1] as follows. Proposition 3.3. Assume that (a) A has enough projective objects; and (b) is a contravariantly finite subcategory of A closed under direct summands, such that P(A). Then for any G D b (A), the following statements are equivalent. (1) G K b ().

Gorenstein Algebras and Recollements 5 (2) There exists i(g) Z, such that Hom D b (A)(G, M[j]) = 0 for any M A and j i(g). (3) There exists a finite set I(G) Z, such that Hom D b (A)(G, M[j]) = 0 for any M A and j / I(G). Furthermore, if A is closed under direct sums, then D b (A) perf = K b (). Proof. The implications (1) (2) and (1) (3) are obvious, (2) (1) Let G D b (A). Then there exists a -quasi-isomorphism G with K,b () by Lemma 2.2, and hence there exists N Z such that H i Hom A (, ) = 0 for any and i N. It follows that H i = 0 for any i N since P(A). We claim that there exists n N such that Im d n. Otherwise, there exists n i(g) such that Im d n /. Put M := Im dn. Then there exists a non-zero epimorphism d n : n M in A, such that d n is identified with the composition of n dn M λ n+1. It induces the following chain map n 1 d n 1 n d n n+1 f M[ n] 0 M 0. d n Notice that Im d n /, so f is not null homotopic. Otherwise, there exists a morphism h : n+1 M such that d n = hdn = hλ d n. Since d n is epic, we have that 1 M = hλ and M is a isomorphic to a direct summand of n+1 in. Since is closed under direct summands by assumption, it follows that M, a contradiction. So Hom K (A)(, M[ n]) 0 and we have Hom D b (A)(G, M[ n]) = Hom D (A)(, M[ n]) = Hom K (A)(, M[ n]) 0, which contradicts the assumption. The claim is proved. Now we have a -quasi-isomorphism n 1 d n 1 n d n n+1 τ n+1 0 Im d n n+1 with τ n+1 K b (). Then G = = τ n+1 in D (A), and it follows that G = τ n+1 K b () in D b (A). Similarly, we get (3) (1). Furthermore, the inclusion K b () D b (A) perf is obvious. onversely, let P D b (A) perf. Then there exists a -quasi-isomorphism P with K,b () by Lemma 2.2, and hence there exists N Z such that H i Hom A (, ) = 0 for any and i N. Since P(A), we have H i = 0 for any i N. We claim that there exists n N such that Im d n. Otherwise, suppose that Im dn / for any n N. Take M := i N Im di and the non-zero morphism d n : n M = Im d n ( i N,i n Im di ),

6 X. Ma, T.W. Zhao, Z.Y. Huang such that d n is identified with the composition of n dn M λ n+1. It induces the following chain map n 1 d n 1 n d n n+1 f M[ n] 0 M 0. Note that Im d n /, so f is not null homotopic. Otherwise, by an argument similar to the above, we have Im d n, a contradiction. So Hom K (A)(, M[ n]) 0 and we have Hom D b (A)(P, M[ n]) = Hom D (A)(, M[ n]) = Hom K (A)(, M[ n]) 0. So there are infinitely many n such that Hom D b (A)(P, M[ n]) 0, which contradicts the assumption that P D b (A) perf. The claim is proved. onsider the following -quasi-isomorphism d n n 1 d n 1 n d n n+1 τ n+1 0 Im d n n+1 with τ n+1 K b (). Then P = = τ n+1 in D (A), and so P = τ n+1 K b () in D b (A). It follows that P K b (). The proof is finished. Because G() is closed under direct summands by [23, Theorem 4.6(2)], the following is an immediate consequence of Proposition 3.3. orollary 3.4. Assume that (a) A has enough projective objects; and (b) G() is a contravariantly finite subcategory of A and P(A). Then for any G DG() b (A), the following statements are equivalent. (1) G K b (G()). (2) There exists i(g) Z, such that Hom D b G() (A) (G, M[j]) = 0 for any M A and j i(g). (3) There exists a finite set I(G) Z, such that Hom D b G() (A) (G, M[j]) = 0 for any M A and j / I(G). Furthermore, if A is closed under direct sums, then DG() b (A) perf = K b (G()). Let D and D be triangulated subcategories of triangulated categories D and D respectively. Recall from [2] that a triangle functor F : D D is said to restrict to D if F restricts to a triangle functor D D. Lemma 3.5. Let F : D D be a triangle functor of triangulated categories. If F admits a right adjoint G, then F restricts to D perf. Proof. Let X D perf. For any Y D, we have Hom D (F X, Y [i]) = Hom D (X, GY [i]) = Hom D (X, (GY )[i]). By the definition of perfect objects, we have F X D perf, and hence F restricts to D perf.

Gorenstein Algebras and Recollements 7 Proposition 3.6. Assume that (1) both A and A are abelian categories having enough projective objects and closed under direct sums. (2) and are contravariantly finite subcategories of A and A closed under direct summands respectively, such that P(A) and P(A ). If F : D b (A) Db (A ) is a triangle functor admitting a right adjoint G, then F restricts to K b (). Proof. It follows from Proposition 3.3 and Lemma 3.5. Our main result is the following Theorem 3.7. Assume that (1) A, A and A are abelian categories having enough projective objects and closed under direct sums; and (2), and are contravariantly finite subcategories of A, A and A closed under direct summands respectively, such that P(A), P(A ) and P(A ). If D b (A) admits a recollement D b i (A ) i D b (A) then D b -sg (A) = 0 if and only if Db -sg (A ) = 0 = D b -sg (A ). j! D b (A ), Proof. By assumption, we have D b (A) = K b (). Let X, Y D b (A ). Since i is fully faithful and i X D b (A)( = K b ()), it follows from Proposition 3.3 that Hom D b (A )(X, Y [i]) = Hom D b (A)(i X, i Y [i]) = Hom D b (A)(i X, (i Y )[i]) = 0 except for finitely many i Z and so X K b ( ). Thus K b ( ) = D b (A ) and D b -sg (A ) = 0. Similarly, we have D b -sg (A ) = 0. onversely, let X D b (A). Then there exists a triangle j! X X i i X (j! X)[1]. By assumption, X K b ( ) and i X K b ( ). Since i restricts to K b ( ) and j! restricts to K b ( ) by Proposition 3.6, we have j! X K b () and i i X K b (), and hence X K b (), which implies that K b () = D b (A) and Db -sg (A) = 0. Recall that an artin algebra A is called Gorenstein if its left self-injective dimension id A A and right selfinjective dimension id A op A are finite. We have the following facts: (1) An artin algebra A is Gorenstein if and only if DG(P(Mod b A))-sg (Mod A) = 0 ([8]); (2) G(P(Mod A)) is contravariantly finite in Mod A ([10, Theorem 3.5]). So, by taking A = Mod A and = G(P(Mod A)) in Theorem 3.7, we have the following result, which is a Gorenstein analogue of [32, Lemma 2.1] as well as a big module analogue of [3, Theorem 4.7] and [18, Theorem 3.5]. orollary 3.8. Let A, A and A be artin algebras. If DG(P(Mod b A)) (Mod A) admits a recollement i DG(P(Mod b A )) (Mod A ) i DG(P(Mod b A)) (Mod A) j! DG(P(Mod b A )) (Mod A ), then A is Gorenstein if and only if so are A and A.

8 X. Ma, T.W. Zhao, Z.Y. Huang It is well known that a ring A has finite left global dimension if and only if D b (Mod A) = K b (P(Mod A)) (that is, DP(Mod b A)-sg (Mod A) = 0). So, by taking A = Mod A and = P(Mod A) in Theorem 3.7, we have the following big module analogue of [32, Lemma 2.1]. orollary 3.9. (cf. [25, orollaries 5 and 6]) Let A, A and A be rings. If D b (Mod A) admits a recollement D b (Mod A i ) i D b j! (Mod A) j D b (Mod A ), then A has finite left global dimension if and only if so do A and A. We remark that the dual counterparts of all results in this section also hold true. 4 haracterizing Gorenstein Algebras In this section, A is an artin algebra. Lemma 4.1. We have (1) For any X K,G(P(mod A))b (G(P(mod A))), the following statements are equivalent. (1.1) X K b (G(P(mod A))). (1.2) For any Y K,G(P(mod A))b (G(P(mod A))), Hom K,G(P(mod A))b (G(P(mod A)))(X, Y [i]) = 0 except for finitely many i Z. (2) For any G K +,G(I(mod A))b (G(I(mod A))), the following statements are equivalent. (2.1) G K b (G(I(mod A))). (2.2) for any Y K +,G(I(mod A))b (GI(mod A)), except for finitely many i Z. Hom K +,G(I(mod A))b (G(I(mod A)))(Y [i], G) = 0 Proof. The first assertion is [3, Lemma 4.5], and the second one is its dual. ompare the following result with the last assertion in orollary 3.4 and its dual counterpart. Proposition 4.2. Let A be virtually Gorenstein. Then we have (1) D b G(P(mod A)) (mod A) perf = K b (G(P(mod A))). (2) co-d b G(I(mod A)) (mod A) coperf = K b (G(I(mod A))). Proof. (1) Let A be virtually Gorenstein. It follows from [12, Theorem 5] that G(P(mod A)) is contravariantly finite in mod A. Then by Lemma 2.2, we have D b G(P(mod A)) (mod A) perf = K,G(P(mod A))b (G(P(mod A))). Now the assertion follows from Lemma 4.1(1). (2) It is dual to (1). We have the following easy observation. Lemma 4.3. The following statements are equivalent. (1) A is Gorenstein.

Gorenstein Algebras and Recollements 9 (2) Any object in G(P(Mod A)) has finite G(I(Mod A))-codimension and any object in G(I(Mod A)) has finite G(P(Mod A))-dimension. (3) Any object in G(P(mod A)) has finite G(I(mod A))-codimension and any object in G(I(mod A)) has finite G(P(mod A))-dimension. Proof. (1) (2) + (3) Let A be Gorenstein. Then any object in Mod A (resp. mod A) has finite G(P(Mod A))-dimension (resp. G(P(mod A))-dimension) by [17, Theorem 11.5.1] (resp. [22, Theorem]). Dually, any object in Mod A (resp. mod A) has finite G(I(Mod A))-dimension (resp. G(I(mod A))- dimension). (2) (1) (resp. (3) (1)) By assumption, we have that D(A A ) has finite G(P(Mod A))-dimension (resp. G(P(mod A))-dimension). So D(A A ) has finite projective dimension by [24, orollary 3.12], and hence id A op A <. Dually, because A A has finite G(I(Mod A))-dimension (resp. G(I(mod A))- dimension) by assumption, we have id A A < by [24, orollary 4.12]. Thus A is Gorenstein. Lemma 4.4. Let A be virtually Gorenstein. Then there exist triangle equivalences DG(P(mod b A)) (mod A) = co-dg(i(mod b A)) (mod A), DG(P(Mod b A)) (Mod A) = co-dg(i(mod b A)) (Mod A). Proof. The first equivalence follows from [3, Theorem 6.3]. By [33, Proposition 3.2], we have that (G(P(Mod A)), G(I(Mod A))) is a balanced pair. Then by [14, Proposition 2.2], we have that an exact sequence is G(P(Mod A))-acyclic if and only if it is G(I(Mod A))- coacyclic, which yields the second equivalence. As a consequence, we have the following Proposition 4.5. Let A be virtually Gorenstein and either X = Mod A or X = mod A. Then we have (1) The following statements are equivalent. (1.1) Any object in G(P(X )) has finite G(I(X ))-codimension. (1.2) K b (G(P(X ))) K b (G(I(X ))) in DG(P(X b ))(X ). (2) The following statements are equivalent. (2.1) Any object in G(I(X )) has finite G(P(X ))-dimension. (2.2) K b (G(I(X ))) K b (G(P(X ))) in DG(P(X b ))(X ). Proof. (1.1) (1.2) Let X K b (G(P(X )))( DG(P(X b )) (X )). Then X co-db G(I(X ))(X ) by Lemma 4.4. We will prove X K b (G(I(X ))) by induction on the width w(x) of X. Let w(x) = 1. By the dual version of [23, orollary 5.12], there exists a finite coproper G(I(X ))-coresolution 0 X G 0 G 1 G n 1 G n 0 of X, which can be viewed as a G(I(X ))-coquasi-isomorphism 0 X 0 0 0 0 0 G 0 G 1 G n 1 G n 0.

10 X. Ma, T.W. Zhao, Z.Y. Huang It follows that X K b (G(I(X ))). Now suppose w(x) = n 2. From the following diagram X 1 0 X 0 X 1 X n 2 0 0 X 0 X 0 X 1 X n 2 X n 1 0 X 2 0 0 0 0 X n 1 0, we get a triangle X 1 X X 2 X 1 [1] with w(x 1 ) = n 1 and w(x 2 ) = 1. By the induction hypothesis, we have X 1 K b (G(I(X ))) and X 2 K b (G(I(X ))), and hence X K b (G(I(X ))). (1.2) (1.1) Let G G(P(X )), as a stalk complex, be an object in K b (G(P(X ))). Then G K b (G(I(X )))( co-dg(i(x b ))(X )) by (1.2). Then there exists a G(I(X ))-coquasi-isomorphism 0 G 0 0 0 0 0 G 0 G 1 G n 1 G n 0; in particular, 0 G G 0 G 1 G n 0 is exact, and hence G has finite G(I(X ))- codimension. Dually, we get (2.1) (2.2). Now we are in a position to prove the following Theorem 4.6. For a virtually Gorenstein algebra A, the following statements are equivalent. (1) A is Gorenstein. (2) K b (G(P(Mod A))) = K b (G(I(Mod A))) in DG(P(Mod b A)) (Mod A). (3) DG(P(Mod b A)) (Mod A) perf = co-dg(i(mod b A)) (Mod A) coperf in DG(P(mod b A)) (Mod A). (4) K b (G(P(mod A))) = K b (G(I(mod A))) in DG(P(mod b A)) (mod A). (5) DG(P(mod b A)) (mod A) perf = co-dg(i(mod b A)) (mod A) coperf in DG(P(mod b A)) (mod A). Proof. By Lemma 4.3 and Proposition 4.5, we have (2) (1) (4). By orollary 3.4 and its dual counterpart, we have DG(P(Mod b A)) (Mod A) perf = K b (G(P(Mod A))) and co-dg(i(mod b A)) (Mod A) coperf = K b (G(I(Mod A))). So the assertion (2) (3) follows. The assertion (4) (5) follows from Proposition 4.2.

Gorenstein Algebras and Recollements 11 In summary, if A is Gorenstein, then we have the following diagram of identifications DG(P(Mod b A)) (Mod A) Theorem 4.6(3) perf co-dg(i(mod b A)) (Mod A) coperf orollary 3.4 K b (G(P(Mod A))) K b (G(P(mod A))) Theorem 4.6(2) Theorem 4.6(4) K b (G(I(Mod A))) K b (G(I(mod A))) The dual of orollary 3.4 Proposition 4.2(1) D b G(P(mod A)) (mod A) perf Proposition 4.2(2) Theorem 4.6(5) co-d b G(I(mod A)) (mod A) coperf. Acknowledgement. This work was partially supported by NSF (No. 11571164), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant Nos. KYX17 0019 and KYZ- Z16 0034), and Nanjing University Innovation and reative Program for PhD candidate (No. 2016011). The authors thank the referee for the useful suggestions, and the first author thanks Javad Asadollahi, Payam Bahiraei and Biao Ma for their discussions. References [1] L. Angeleri Hügel, S. König and. H. Liu, Recollements and tilting objects, J. Pure Appl. Algebra 215 (2011), 420 438. [2] L. Angeleri Hügel, S. König and. H. Liu and D. Yang, Ladders and simplicity of derived module categories, J. Algebra 472 (2017), 15 66. [3] J. Asadollahi, P. Bahiraei, R. Hafezi and R. Vahed, On relative derived categories, omm. Algebra 44 (2016), 5454 5477. [4] J. Asadollahi, R. Hafezi and R. Vahed, Gorenstein derived equivalences and their invariants, J. Pure Appl. Algebra 218 (2014), 888 903. [5] M. Auslander and R.-O. Buchweitz, The homological theory of maximal ohen-macaulay approximations, Mém. Soc. Math. France 38 (1989), 5 37. [6] M. Auslander and I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), 111 152. [7] L. L. Avramov and A. Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. 85 (2002), 393 440. [8] Y. H. Bao, X. N. Du, and Z. B. Zhao, Gorenstein singularity categories, J. Algebra 428 (2015), 122 137. [9] A. A. Beĭlinson, J. Bernstein and P. Deligne, Faisceaux pervers, Analysis and Topology on Singular Spaces I (Luminy, 1981), Astérisque 100, Soc. Math. France, Paris, 1982, pp.5 171.

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