Applications of balanced pairs

Transcription

1 SCIENCE CHINA Mathematics. ARTICLES. May 2016 Vol.59 No.5: doi: /s Applications o balanced pairs LI HuanHuan, WANG JunFu & HUANG ZhaoYong Department o Mathematics, Nanjing University, Nanjing , China lihuanhuan0416@163.com, wangjunu05@126.com, huangzy@nju.edu.cn Received November 21, 2014; accepted July 13, 2015; published online November 24, 2015 Abstract Let (X, Y ) be a balanced pair in an abelian category. We irst introduce the notion o cotorsion pairs relative to (X, Y ), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perect. We prove that i the X -resolution dimension o Y (resp. Y -coresolution dimension o X ) is inite, then the bounded homotopy category o Y (resp. X ) is contained in that o X (resp. Y ). As a consequence, we get that the right X -singularity category coincides with the let Y -singularity category i the X -resolution dimension o Y and the Y -coresolution dimension o X are inite. Keywords balanced pairs, relative cotorsion pairs, relative derived categories, relative singularity categories, relative (co)resolution dimension MSC(2010) 16G25, 18G10, 18G20 Citation: Li H H, Wang J F, Huang Z Y. Applications o balanced pairs. Sci China Math, 2016, 59: , doi: /s Introduction Cartan and Eilenberg [4] introduced the notions o right and let balanced unctors. Then Enochs and Jenda[9]generalizedthemtorelativehomologicalalgebraasollows. Let C, D and E beabeliancategories and T(, ) : C D E be an additive unctor contravariant in the irst variable and covariant in the second. Then T is called right balanced by F G i or any M C, there exists a T(, G)-exact complex F 1 F 0 M 0 with each F i F, and or any N D, there exists a T(F, )-exact complex 0 N G 0 G 1 with each G i G. They showed that i T is right balanced by F G, and i F M is a T(, G)-exact complex and N G is a T(F, )-exact complex, then the complexes T(F,N) and T(M,G ) have isomorphic homology. There are many examples o right balanced unctors in the module category when we regard T as Hom, see [10, Chapter 8]. Recently, Chen [5] introduced the notion o balanced pairs o additive subcategories in an abelian category. Let A be an abelian category with enough projectives and injectives. We use P(A) and I(A) to denote the ull subcategories o A consisting o projectives and injectives respectively. It is known that the pair (P(A), I(A)) is a balanced pair, which is called the classical balanced pair. Chen [5] showed that or a balanced pair (X, Y ) o A, it inherits some nice properties rom the classical one. The notion o cotorsion pairs was irst introduced by Salce [20], and it has been deeply studied in homological algebra, representation theory and triangulated categories in recent years, see [11, 14 17], and so on. In particular, Hovey[14] established a connection between cotorsion pairs in abelian categories and model category theory. In classical homological algebra, the deinition o the cotorsion pair is based Corresponding author c Science China Press and Springer-Verlag Berlin Heidelberg 2015 math.scichina.com link.springer.com

2 862 Li H H et al. Sci China Math May 2016 Vol. 59 No.5 on the unctor Ext i A (, ). The advantage is that this unctor is independent o the choices o the projective resolutions o the irst variable, and also independent o the choices o the injective resolutions o the second variable. In other words, the cotorsion pair is essentially based on the balanced pair (P(A), I(A)). Based on these backgrounds mentioned above, it is natural or us to introduce and study cotorsion pairs relative to balanced pairs, and we show that relative cotorsion pairs share many nice properties o the classical one. This paper is organized as ollows. In Section 2, we give some terminologies and some preliminary results. In Section 3, or an abelian category A, we introduce the notion o cotorsion pairs relative to a given balanced pair (X, Y ). Similar to the classical case, we also introduce the notions o complete, hereditary and perect cotorsion pairs relative to (X, Y ), and obtain some equivalent characterizations or the cotorsion pair relative to (X, Y ) being complete, hereditary and perect, respectively. In Section 4, or a given balanced pair (X, Y ) o the abelian category A, we introduce the notions o the right X -derived category DRX (A) and the let Y -derived category D LY (A) o A or {blank,,+,b}. We show that in the bounded case, they are actually the same, and we denote both by D (A). b Let (X, Y ) be an admissible balanced pair. We give some criteria or computing the X -resolution dimension and the Y -coresolution dimension o an object in A in terms o the vanishing o relative cohomology groups. Moreover, we show that i the X -resolution dimension o Y (resp. Y - coresolution dimension o X ) is inite, then the bounded homotopy categoryo Y (resp. X ) is contained in that o X (resp. Y ). This generalizes a classical result o Happel. As a consequence, we get that the right X -singularity category coincides with the let Y -singularity category i the X -resolution dimension o Y and the Y -coresolution dimension o X are inite. 2 Preliminaries Throughout this paper, A is an abelian category. For a subcategory o A we mean a ull additive subcategory closed under isomorphisms and direct summands. We use P(A) and I(A) to denote the subcategories o A consisting o projective and injective objects, respectively. We use C(A) to denote the category o complexes o objects in A, K (A) to denote the homotopy category o A, and D (A) to denote the usual derived category by inverting the quasi-isomorphisms in K (A), where {blank,,+,b}. Let X := X dn 1 n 1 X X n dn X X n+1 be a complex in C(A) and : X Y a cochain map in C(A). We use Con() to denote the mapping cone o. Recall that X is called acyclic (or exact) i H i (X ) = 0 or any i Z (the ring o integers), and is called a quasi-isomorphism i H i () is an isomorphism or any i Z. We have that is a quasi-isomorphism i and only i Con() is acyclic. Deinition 2.1. (1) (See [2]) Let X Y be subcategories o A. A morphism : X Y in A with X X and Y Y is called a right X -approximation o Y i or any morphism g : X Y in A with X X, there exists a morphism h : X X such that the ollowing diagram commutes: X h X Y. I any endomorphism s : X X is an automorphism whenever = s, then is called right minimal. I each object in Y has a right X -approximation, then X is called contravariantly inite in Y. Dually, the notions o let X -approximations, let minimal morphisms and covariantly inite subcategories are deined. (2) (See [5]) A contravariantly inite subcategory X o A is called admissible i each right X - approximation is epic. Dually, the notion o coadmissible subcategories is deined. g

3 Li H H et al. Sci China Math May 2016 Vol. 59 No Deinition 2.2. (1) (See [5]) Given a subcategory X o A. A complex A in C(A) is called right (resp. let) X -acyclic i the complex Hom A (X,A ) (resp. Hom A (A,X) ) is acyclic or any X X. A cochain map : A B in C(A) is said to be right (resp. let) X -quasi-isomorphism i the cochain (resp. chain) map Hom A (X,) (resp. Hom A (,X)) is a quasi-isomorphism or any X X. It is equivalent to that Con() is right (resp. let) X -acyclic. (2) (See [5, 10]) Given a contravariantly inite subcategory X o A and an object M A. An X -resolution o M is a complex X 2 d 2 X 1 d 1 X 0 ε M 0 in A with each X i X such that it is right X -acyclic. Usually we denote the complex by X ε M or short, where X := X 2 d 2 X 1 d 1 X 0 0 is the deleted X -resolution o M. The X -resolution dimension X -res.dimm o M is deined to be the minimal integer n 0 such that there exists an X -resolution: 0 X n X 1 X 0 M 0. I no such an integer exists, we set X -res.dimm =. The global X -resolution dimension X -res.dima o A is deined to be the supreme o the X -resolution dimensions o all objects in A. Dually, i X is a covariantly inite subcategory o A, then the notions o X -coresolutions, X - coresolution dimensions and the global X -coresolution dimension are deined. Lemma 2.3. Let X be a subcategory o A and let 0 L M g N 0 (2.1) be an acyclic complex. (1) I (2.1) is right X -acyclic, then or any morphisms N α N and L s L, we have the ollowing pull-back diagram with the upper row right X -acyclic: 0 L M g N 0 0 L β α M g N 0, and the ollowing push-out diagram with the bottom row right X -acyclic: 0 L M g N 0 s 0 L t M g N 0. (2) I (2.1) is let X -acyclic, then or any morphisms N α N and L s L, we have the ollowing pull-back diagram with the upper row let X -acyclic: 0 L M g N 0 0 L β α M g N 0, and the ollowing push-out diagram with the bottom row let X -acyclic: 0 L M g N 0 s 0 L t M g N 0.

4 864 Li H H et al. Sci China Math May 2016 Vol. 59 No.5 Proo. (1) Because the sequence (2.1) is right X -acyclic by assumption, or any morphism h : X N with X X there exists a morphism i : X M such that αh = gi. Since the right square in the irst diagram is a pull-back diagram, there exists a morphism φ : X M such that h = g φ. It implies that the upper row in this diagram is right X -acyclic. Also because the sequence (2.1) is right X -acyclic acyclic, or any morphism h : X N with X X there exists a morphism i : X M such that h = gi = g ti. It implies that the bottom row in the second diagram is right X -acyclic. (2) It is dual to (1). Lemma 2.4. Let A be a complex in C(A). Then A is right X -acyclic i and only i the complex Hom A (X,A ) is acyclic or any X K (X ). Proo. See [8, Lemma 2.4]. Lemma 2.5. (1) Let X be a complex in K (X ) and let : A X be a right X -quasi-isomorphism in C(A). Then there exists a cochain map g : X A such that g is homotopic to id X. (2) Any right X -quasi-isomorphism between two complexes in K (X ) is a homotopy equivalence. Proo. (1) Consider the distinguished triangle: A X Con() A [1] in K(A) with Con() right X -acyclic. By applying the unctor Hom K(A) (X, ) to it, we get an exact sequence: Hom K(A) (X,A ) Hom K(A)(X,) Hom K(A) (X,X ) Hom K(A) (X,Con()). It ollows rom Lemma 2.4 that Hom K(A) (X,Con()) = H 0 Hom A (X,Con()) = 0. So there exists a cochain map g : X A such that g is homotopic to id X. (2) It is a consequence o (1). 3 Cotorsion pairs relative to balanced pairs Deinition 3.1 (See [5,10]). A pair (X, Y ) o subcategories o A is called a balanced pair i the ollowing conditions are satisied: (1) X is contravariantly inite in A and Y is covariantly inite in A. (2) For any object M A, there exists an X -resolution X M o M such that it is let Y -acyclic. (3) For any object N A, there exists a Y -coresolution N Y o N such that it is right X -acyclic. We list some examples o balanced pairs as ollows. Example 3.2. (1) Recall that A is said to have enough projectives (resp. enough injectives) i or any M A, there exists an epimorphism P M 0 (resp. a monomorphism 0 M I) with P (resp. I) in P(A) (resp. I(A)). In case or A having enough projectives and injectives, it is well known that the pair (P(A), I(A)) is a balanced pair. We call it the classical balanced pair. (2) (See [10, Example 8.3.2]) Let R be a ring and ModR the category o let R-modules, and let PP(R) and PI(R) be the subcategories o ModR consisting o pure projective modules and pure injective modules respectively. Then (PP(R), PI(R)) is a balanced pair in Mod R. (3) (See [10, Theorem ]) Let R be an n-gorenstein ring (that is, R is a let and right Noetherian ring with let and right sel-injective dimensions at most n), and let GProjR and GInjR be the subcategories o Mod R consisting o Gorenstein projective and Gorenstein injective modules respectively. Then the pair (GProjR, GInjR) is a balanced pair in ModR. Let X (resp. Y ) be a contravariantly inite (resp. covariantly inite) subcategory o A. Then the pair (X, Y ) is a balanced pair i and only i the class o right X -acyclic complexes coincides with that o let Y -acyclic complexes (see [5, Proposition 2.2]). In what ollows, we call a complex -acyclic i it is both right X -acyclic and let Y -acyclic.

5 Li H H et al. Sci China Math May 2016 Vol. 59 No Deinition 3.3 (See [10, Deinition 7.1.2]). Let A have enough projectives and injectives. A pair (C, D) o subcategories o A is called a cotorsion pair i C = D and D = C, where D = {C A Ext 1 A(C,D) = 0 or any D D} and C = {D A Ext 1 A(C,D) = 0 or any C C}. Notice that the unctor Ext 1 A(, ) is based on the classical balanced pair (P(A), I(A)), it induces an isomorphism o cohomology groups whether we take a projective resolution o the irst variable or take an injective coresolution o the second variable. From this viewpoint we may say that the cotorsion pair deined above is a cotorsion pair relative to the balanced pair (P(A), I(A)). Let (X, Y ) be a balanced pair and M,N A. Choose an X -resolution X M o M and a Y -coresolution N Y o N. We get two cohomological groups Ext i X (M,N) := Hi (Hom A (X,N)) and Ext i Y(M,N) := H i (Hom A (M,Y )) or any i Z. They are independent o the choices o the X - resolutions om and the Y -coresolutionso N respectively. For any i Z, there exists an isomorphism o abelian groups Ext i X (M,N) = Ext i Y (M,N) (see [10]). We denote both abelian groups by Exti (M,N). Motivated by the above argument, we introduce the ollowing: Deinition 3.4. Let (X, Y ) be a balanced pair. A pair (C, D) o subcategories o A is called a cotorsion pair relative to (X, Y ) i C= D and D=C, where D = {C A Ext 1 (C,D) = 0 or any D D} and C = {D A Ext 1 (C,D) = 0 or any C C}. In the rest o this section, we ix a balanced pair (X, Y ) and a cotorsion pair (C,D) relative to (X, Y ) in A. Proposition 3.5. Y Y. For any A A, we have Ext 1 (X,A) = 0 = Ext 1 (A,Y) or any X X and Proo. It is straightorward. Deinition 3.6. (1) Let E be a subcategory o A. E is said to be closed under -extensions i or any -acyclic complex 0 L M N 0 in A, L,N E implies M E; E is said to be closed under -epimorphisms i or any -acyclic complex 0 L M N 0 in A, M,N E implies L E; E is said to be closed under -monomorphisms i or any -acyclic complex 0 L M N 0 in A, L,M E implies N E. (2) A subcategory E o A is called X -resolving i X E and E is closed under -extensions and -epimorphisms; and E is called Y -coresolving i Y E and E is closed under -extensions and - monomorphisms. Proposition 3.7. (1) X C and Y D. (2) Both C and D are closed under -extensions. Proo. (1) It is trivial. (2) Let 0 L M N 0 be a -acyclic complex in A with L,N C. By [10, Theorem 8.2.3], or any D D we have an acyclic complex Ext 1 (N,D) Ext1 (M,D) Ext1 (L,D). Then we have Ext 1 (N,D) = 0 = Ext1 (L,D). So Ext1 (M,D) = 0 and M C. Thus C is closed under -extensions. Similarly, we have that D is closed under -extensions. The ollowing result is a relative version o [2, Lemmas 3.1 and 3.2]. Theorem 3.8. The ollowing statements are equivalent: (1) C is X -resolving. (2) D is Y -coresolving. (3) Ext 1 (C,D) = 0 or any C C and D D. In this case, (C,D) is called hereditary.

6 866 Li H H et al. Sci China Math May 2016 Vol. 59 No.5 Proo. (1) (3). Let C C and D D, and let 0 K X C 0 be a -acyclic complex in A with X X. Then X C by Proposition 3.7(1), and so K C by (1). Hence By [10, Theorem 8.2.3], we have an exact sequence: Ext 1 (K,D) = 0. Ext 1 (X,D) Ext 1 (K,D) Ext 2 (C,D) Ext 2 (X,D). Since Ext 1 (X,D) = 0 = Ext 2 (X,D) by Proposition 3.5, we have Ext 2 (C,D) = Ext 1 (K,D) = 0. We get Ext 1 (C,D) = 0 inductively. (3) (1). Let 0 L M N 0 be a -acyclic complex in A with M,N C. By [10, Theorem 8.2.3], or any D D we have the ollowing exact sequence: Ext 1 (M,D) Ext1 (L,D) Ext2 (N,D). Because Ext 1 (M,D) = 0 = Ext2 (N,D) by assumption, we have Ext 1 (L,D) = 0 and L C. Now the assertion ollows rom Proposition 3.7. Dually, we get (2) (3). Deinition 3.9 (See [10, Deinition 7.1.5]). M A, there exists a -acyclic complex (1) (C, D) is said to have enough projectives i or any 0 D C M 0 in A with C C and D D; and it is said to have enough injectives i or any M A, there exists a -acyclic complex 0 M D C 0 in A with C C and D D. (2) I (C, D) has enough projectives and enough injectives, then it is called complete. Proposition admissible. Proo. See [5, Corollary 2.3]. X is admissible i and only i Y is coadmissible. In this case, (X, Y ) is called It is obvious that i (X, Y ) is admissible, then each -acyclic complex is acyclic. Theorem enough injectives. I (X, Y ) is admissible, then (C,D) has enough projectives i and only i it has Proo. We only show the i part, and the only i part ollows dually. Assume that (C, D) has enough injectives and M A. Choose a -acyclic complex 0 K X M 0 in A with X X. Since (C, D) has enough injectives, there exists a -acyclic complex 0 K D C 0

7 Li H H et al. Sci China Math May 2016 Vol. 59 No in A with C C and D D. Because (X, Y ) is admissible, each -acyclic complex is acyclic. So we have the ollowing push-out diagram with acyclic columns and rows: K X 0 D E M 0 M 0 C C 0 0. Then all o columns and rowsare -acyclicby Lemma 2.3. Since X,C C, it ollows rom Proposition3.7 that E C. The assertion ollows. Lemma Let (X, Y ) be admissible and E a subcategory o A which is closed under -extensions. (1) I ϕ : E M is a minimal right E-approximation and 0 S i P π G 0 E ϕ M is a commutative diagram with G E such that the upper row is -acyclic, then there exists a morphism α : P E such that = αi and θ = ϕα. (2) I ψ : M E is a minimal let E-approximation and θ M ψ E 0 F θ i Q π K 0 is a commutative diagram with F E such that the bottom row is -acyclic, then there exists a morphism α : E Q such that θ = αψ and = πα. Proo. (1) Consider the ollowing push-out diagram: 0 S i P π G 0 0 E j X G 0. Because the irst row is -acyclic by assumption, it ollows rom Lemma 2.3 that the bottom row is also - acyclic. Since E,G E, we have X E. By the universal property o push-outs there exists a morphism h : X M such that ϕ = hj and θ = hk. Because ϕ : E M is a minimal right E-approximation by assumption, there exists a morphism g : X E such that h = ϕg. Thus ϕ = hj = ϕgj, which implies that gj : E E is an automorphism. We may assume gj = id E. Then by letting α = gk, we have j = ki = jgki = jαi. Since j is a monomorphism, = αi. It ollows rom θ = hk and h = ϕg that θ = hk = ϕgk = ϕα, we complete the proo. (2) It is dual to (1). k

8 868 Li H H et al. Sci China Math May 2016 Vol. 59 No.5 The ollowing result is a relative version o the Wakamatsu s lemma. Proposition Let (X, Y ) be admissible and E a subcategory o A which is closed under - extensions. Then we have the ollowing: (1) The kernel o every minimal right E-approximation is in E. (2) The cokernel o every minimal let E-approximation is in E. Proo. (1) Let ϕ : E M be a minimal right E-approximation o an object M in A, and let K := Kerϕ and i : K E be the inclusion. Because X is contravariantly inite in A, or any E E there exists a -acyclic complex, 0 S X E 0 in A with X X. By applying the unctor Hom A (,K) we get an exact sequence: Hom A (X,K) Hom A (S,K) Ext 1 (E,K) 0. For any morphism : S K, it ollows rom Lemma 3.12 that there exists a morphism g : X E such that the ollowing diagram is commutative. Then i S X g E ϕ M Img Kerϕ = K. 0 So the map Hom A (X,K) Hom A (S,K) is epic and Ext 1 (E,K) = 0. (2) It is dual to (1). Deinition (C, D) is called perect i every object o A has a minimal right C-approximation and a minimal let D-approximation. Let (X, Y ) be admissible. I (C, D) is perect, then it is complete by Proposition The ollowing result is a relative version o [11, Theorem 3.8]. Theorem Let (X, Y ) be admissible and (C, D) a hereditary cotorsion pair. Then the ollowing statements are equivalent: (1) (C, D) is perect. (2) Every object o A has a minimal right C-approximation and every object o C has a minimal let D-approximation. (3) Every object o A has a minimal let D-approximation and every object o D has a minimal right C-approximation. Proo. Both (1) (2) and (1) (3) are trivial. In the ollowing we only prove (2) (1), and (3) (1) ollows dually. Let ϕ : C M be a minimal right C-approximationoan object M in A. Since (X, Y ) is admissible, by Proposition 3.13(1) that there exists a -acyclic complex, 0 D i C ϕ M 0 in A with D D. Let ψ : C D be a minimal let D-approximationoC. Then byproposition3.13(2), we get a -acyclic complex 0 C ψ D π C 0

9 Li H H et al. Sci China Math May 2016 Vol. 59 No in A with C C. Since C,C C, we have D C D. Consider the ollowing push-out diagram: 0 0 D i C 0 ϕ M 0 0 D i D ψ ϕ X ψ 0 C π C 0 0. By Lemma 2.3 the rightmost column is -acyclic. For any morphism : D Y with Y Y, there exists a morphism g : C Y such that = gi, and then there exists a morphism j : D Y such that g = jψ. It ollows that = gi = jψi = ji and the middle row is -acyclic. Because (C, D) is hereditary by assumption, we have X D. To get the desired assertion, it suices to show ψ : M X is let minimal. Let h : X X satisying ψ = hψ. By applying the unctor Hom A (D, ) to the middle row, we have a morphism h : D D such that the ollowing diagram D ϕ X h h D ϕ X is commutative. Hence we have the ollowing commutative diagram: D ϕ X h D ϕ ψ X M. h ψ M Note that the diagram C ϕ M ψ D ϕ X is both a push-out diagram and a pull-back diagram. Then there exists morphism h : C C such that the ollowing diagram D D C C X X ψ M M is commutative. Since ϕ : C M is right minimal, h : C C is an automorphism. Then it ollows rom the let minimality o ψ : C D that h : D D is also an automorphism. It implies that h : X X is an automorphism and ψ : M X is let minimal.

10 870 Li H H et al. Sci China Math May 2016 Vol. 59 No.5 4 Derived categories relative to balanced pairs Let X be a subcategoryo A. It is known that K (A) is a triangulated categoryor {blank,,+,b}. Denote by KRX ac (A) (resp. K LX ac (A)) the ull triangulated subcategory o K (A) consisting o right X -acyclic (resp. let X -acyclic) complexes. Both o them are thick subcategories because they are closed under direct summands. Denote by RX (resp. LX ) the class o all right (resp. let) X -quasi-isomorphisms in K (A). Then a cochain map is a right (resp. let) X -quasi-isomorphism i and only i its mapping cone is right (resp. let) X -acyclic. Thus RX (resp. LX ) is the saturated compatible multiplicative system determined by KRX ac (A) (resp. K LX ac (A)). Deinition 4.1 (See [21]). The Verdier quotient category DRX (A) := K (A)/KRX-ac (A) is called the right X -derived category o A, where {blank,,+,b}. The let X -derived category DLX (A) o A is deined dually. Example 4.2. Let A have enough projectives. (1) I X = P(A), then D RX (A) is the usual derived category D (A). (2) I X = G(A) (the subcategory o A consisting o Gorenstein projective objects), then D RX (A) is the Gorenstein derived category D gp(a) deined in [12]. The ollowing two results are cited rom [1]. Proposition 4.3 (See [1]). (1) D RX (A) is a triangulated subcategory o D RX(A), and DRX b (A) is a triangulated subcategory o D RX (A). (2) For any X K (X ) and C C(A), there exists an isomorphism o abelian groups: Hom K(A) (X,C ) = Hom DRX (A)(X,C ). (3) Let X A be admissible. Then the composition unctor A K b (A) DRX b (A) is ully aithul, where both unctors are canonical ones. and Set K,RXb (X ) := {X K (X ) there exists n Z such that H i (Hom A (X,X )) = 0 or any X X and i n}, K +,LXb (X ) := {X K + (X ) there exists n Z such that H i (Hom A (X,X)) = 0 or any X X and i n}. Proposition 4.4 (See [1, Theorem 3.3]). I X is a contravariantly inite subcategory o A, then we have a triangle-equivalence K,RXb (X ) DRX b (A). As consequences o Propositions 4.4 and 4.3, we have the ollowing two results. Proposition 4.5. Let (X, Y ) be a balanced pair in A. Then we have triangle-equivalences: K,RXb (X ) D b RX(A) = D b LY(A) K +,LY b (Y ). Proo. The irst equivalence ollows rom Proposition 4.4, and the last one is its dual. By [5, Proposition 2.2], we have that the class o right X -acyclic complexes coincides with that o let Y -acyclic complexes i (X, Y ) is a balanced pair. Then KRX-ac b (A) coincides with Kb LY -ac(a), and so we have DRX b (A) = Db LY (A). For a balanced pair (X, Y ) in A, we call DRX b (A) and Db LY (A) the relative bounded derived category relativeto (X, Y ), and denote them by D b (A). The ollowingresult means that the relativecohomology group Ext i (M,N) may be computed in the relative bounded derived category D (A). b Proposition 4.6. Let (X, Y ) be a balanced pair in A. Then or any M,N A and i 1, there exists an isomorphism o abelian groups: Ext i (M,N) = Hom D b (A)(M,N[i]).

11 Li H H et al. Sci China Math May 2016 Vol. 59 No Proo. Let ε : X M M be a X -resolution o M. View M as a stalk complex concentrated in degree zero. Note that ε is a right X -quasi-isomorphism. So M = X M in D RX(A). Since X M K (X ), by Proposition 4.3 we have isomorphisms o abelian groups: Ext i (M,N) = H i Hom A (X M,N) = Hom K(A) (X M,N[i]) = Hom DRX (A)(X M,N[i]) = Hom DRX (A)(M,N[i]) = Hom D b RX (A)(M,N[i]) = Hom D b (A)(M,N[i]). Given a balanced pair (X, Y ) in A, rom the viewpoint o relative derived category relative to (X, Y ), the X -resolution o an object M A is exactly an isomorphism X M M in Db (A), where X M K (X ) with components vanish in the positive degrees, while the Y -coresolution o an object N A is exactly an isomorphism N Y N in Db (A), where Y N K+ (Y ) with components vanish in the negative degrees. In the ollowing result, we give some criteria or computing the X -resolution dimension o an object in A in terms o the vanishing o relative cohomology groups. Theorem 4.7. Let (X, Y ) be an admissible balanced pair in A. Then the ollowing statements are equivalent or any M A and n 0: (1) X -res.dimm n. (2) Ext n+1 (M,N) = 0 or any N A. (3) Ext n+1 (M,N) = 0 or any N A. (4) For any X -resolution X M o M, we have Kerd n+1 X X. Proo. Both (2) (3) and (4) (1) are trivial. (1) (2) Let 0 X n X n+1 X 0 M 0 be an X -resolution o M. Then Hom A (X i,n) =0 or any N A and i n + 1 and the assertion ollows. (3) (4) Let X d n n X X n+1 X 0 M 0 be an X -resolution o M. Then we have a -acyclic sequence: Since Ext n+1 0 Kerd n X X n Kerd n+1 X 0. (4.1) (M,Kerd n X ) = 0, by the dimension shiting we have Ext 1 (Kerd n+1 X,Kerd n X ) = Ext n+1 (M,Kerd n X ) = 0. It ollows rom [10, Theorem 8.2.3] that (4.1) splits. So Kerd n+1 X X n and Kerd n+1 X X. Dually, we have the ollowing theorem. is isomorphic to a direct summand o Theorem 4.8. Let (X, Y ) be an admissible balanced pair in A. Then the ollowing statements are equivalent or any N A and n 0: (1) Y -cores.dimn n. (2) Ext n+1 (M,N) = 0 or any M A. (3) Ext n+1 (M,N) = 0 or any M A. (4) For any Y -coresolution N Y o N, we have Imd n 1 Y Y. As an immediate consequence o Theorems 4.7 and 4.8, we have the ollowing corollary. Corollary 4.9 (See [5, Corollary 2.5]). X -res.dima = Y -cores.dima. and Set Let (X, Y ) be an admissible balanced pair in A. Then X -res.dimy := sup{x -res.dimy Y Y }, Y -cores.dimx := sup{y -cores.dimx X X }.

12 872 Li H H et al. Sci China Math May 2016 Vol. 59 No.5 Theorem For a balanced pair (X, Y ) in A, in D b (A) we have (1) I X -res.dim Y <, then K b (Y ) K b (X ). (2) I Y -cores.dimx <, then K b (X ) K b (Y ). Proo. (1) It suices to show that or any Y K b (Y ), there exists a right X -quasi-isomorphism X Y Y with X Y Kb (X ). We proceed by induction on the width ω(y ) (:= the cardinal o the set {Y i 0 i Z}) o Y. For the case ω(y )=1, the assertion ollows rom the assumption that X is contravariantly inite and X -res.dimy <. Let ω(y ) 2 with Y j 0 and Y i = 0 or any i < j. Put Y 1 := Y j [ j 1] and Y 2 := σ>j Y. Let g = d j Y [ j 1] where dj Y is the j-th dierential o Y. We have a distinguished triangle Y 1 g Y 2 Y Y 1 [1] in K b (Y ). By the induction hypothesis, there exist right X -quasi-isomorphisms Y1 : X Y 1 Y 1 and Y2 : X Y 2 Y 2 with X Y 1, X Y 2 K b (X ). Then by Lemma 2.4, Y2 induces an isomorphism: Hom K b (A)(X Y 1,X Y 2 ) = Hom K b (A)(X Y 1,Y 2 ). So there exists a morphism : X Y 1 X Y 2, which is unique up to homotopy, such that Y2 = g Y1. Put X Y = Con(). We have the ollowing distinguished triangle: X Y 1 X Y2 X Y X Y 1 [1] in K b (X ). Then there exists a morphism Y : X Y Y such that the ollowing diagram commutes: X Y 1 Y 1 Y1 g X Y 2 Y 2 Y2 X Y X Y 1 [1] Y Y Y 1 [1]. Y1 [1] For any X X and n Z, we have the ollowing commutative diagram with exact rows: (X,X Y 1 ) (X,X Y 2 ) (X,X Y ) (X,X Y 1 [1]) (X,X Y 2 [1]) (X, Y1 ) (X, Y2 ) (X, Y ) (X, Y1 [1]) (X,Y 1 ) (X,Y 2 ) (X,Y ) (X,Y 1 [1]) (X,Y 2 [1]), (X, Y2 [1]) where(x, )denotestheunctorhom K(A) (X,[n]( )). Since Y1 and Y2 areright X -quasi-isomorphisms, we have that (X, Y1 ) and (X, Y2 ) are isomorphisms. So (X, Y ) is also an isomorphism and Y is a right X -quasi-isomorphism. The proo is inished. (2) It is dual to (1). Let A be a inite-dimensional algebra over a ield k. We use moda to denote the category o initely generated let A-modules, and use proja (resp. inja) to denote the ull subcategory o moda consisting o projective (resp. injective) modules. For a module M moda, we use pd A M and id A M to denote the projective and injective dimensions o M, respectively. As an application o Theorem 4.10, we get the ollowing corollary. Corollary 4.11 (See [13]). For a inite-dimensional algebra A over a ield k, in D b (A) we have the ollowing: (1) pd A D(A A ) < i and only i K b (inja) K b (proja). (2) id A A < i and only i K b (proja) K b (inja). (3) A is Gorenstein i and only i K b (proja) = K b (inja).

13 Li H H et al. Sci China Math May 2016 Vol. 59 No Proo. We only prove(1), because(2) is dual to (1), and (3) is an immediate consequenceo(1) and(2). The necessity ollows rom Theorem For the suiciency, since K b (inja) K b (proja) in D b (A), we have D(A A ) K b (proja). Then there exists a quasi-isomorphism Q D(A A ) with Q K b (proja). Let P D(A A ) be the projective resolution o D(A A ) in moda. It ollows that P and Q are homotopy equivalence. Thus P K b (proja) and hence pd A D(A A ) <. Let (X, Y ) be a balanced pair in A. It ollows rom Proposition 4.3 that K b (X ) is a triangulated subcategory o D b (A). Motivated by the deinition o classical singularity categories, we introduce the ollowing deinition. Deinition We call the quotient category D RX-sg (A) := D b (A) /Kb (X ) the right X -singularity category relative to (X, Y ), and call D LY -sg (A) := D b (A) /Kb (Y ) the let Y -singularity category relative to (X, Y ). Let A be a inite-dimensional algebra over a ield k. In the case or X = proja, we have that D b X (A) coincides with the usual bounded derived category D b (A) and D RX-sg (A) is the classical singularity category D sg (A) which is called the stabilized derived category in [3]. For the properties o singularity categories and related topics, we reer to [6,7,13,18,19] and so on. It is known that D sg (A) = 0 i and only i A is o inite global dimension. So D sg (A) measures the homological singularity o the algebra A. Theorem Let (X, Y ) be a balanced pair in A. (1) I X -res.dimy < and Y -cores.dimx <, then D RX-sg (A) = D LY -sg (A). (2) I (X, Y ) is admissible and X -res.dima <, then D RX-sg (A) = 0 = D LY -sg (A). Proo. (1) I X -res.dimy < and Y -cores.dimx <, then it ollows rom Theorem 4.10 that K b (X ) = K b (Y ). So we have D RX-sg (A) = D LY -sg (A). (2) Since (X, Y ) is an admissible balanced pair and X -res.dim A <, it ollows rom Corollary 4.9 that Y -cores.dima <. For the irst equality it suices to show that or any A K b (A), there exists a right X -quasi-isomorphism X A A with X A Kb (X ). By using an induction on the width ω(a ) o A and a similar argument to that in proo o Theorem 4.10, we get the assertion. Dually, we get the second equality. Let A be a inite-dimensional algebra over a ield k. We use GprojA (resp. GinjA) to denote the ull subcategory o mod A consisting o Gorenstein projective (resp. injective) modules. It ollows rom [5, Proposition2.6]that (GprojA, GinjA) is anadmissiblebalancedpairin modawheneveraisgorenstein. Let (X, Y )=(Gproj A, Ginj A). Consider the ollowing quotient categories (see [12]): D RG -sg(a) := D b (moda)/k b (GprojA), D LG -sg(a) := D b (moda)/kb (GinjA). By Theorem 4.13(2), we have the ollowing corollary. Corollary Let A be a Gorenstein algebra. Then D RG -sg(a) = 0 and D LG -sg(a) = 0. Acknowledgements This research was supported by National Natural Science Foundation o China (Grant No ). Reerences 1 Asadollahi J, Haezi R, Vahed R. Gorenstein derived equivalences and their invariants. J Pure Appl Algebra, 2014, 218: Auslander M, Reiten I. Applications o contravariantly inite subcategories. Adv Math, 1991, 86: Buchweitz R O. Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings. Unpublished manuscript, Cartan H, Eilenberg S. Homological Algebra. Princeton: Princeton University Press, Chen X W. Homotopy equivalences induced by balanced pairs. J Algebra, 2010, 324: Chen X W. Relative singularity categories and Gorenstein-projective modules. Math Nachr, 2011, 284:

14 874 Li H H et al. Sci China Math May 2016 Vol. 59 No.5 7 Chen X W, Zhang P. Quotient triangulated categories. Manuscripta Math, 2007, 123: Christensen L W, Frankild A, Holm H. On Gorenstein projective, injective and lat dimensions a unctorial description with applications. J Algebra, 2006, 302: Enochs E E, Jenda O M G. Balanced unctors applied to modules. J Algebra, 1985, 92: Enochs E E, Jenda O M G. Relative Homological Algebra. Berlin-New York: Walter de Gruyter, Enochs E E, Jenda O M G, Torrecillas B, et al. Torsion theory relative to Ext. summary?doi= , Gao N, Zhang P. Gorenstein derived categories. J Algebra, 2010, 323: Happel D. On Gorenstein algebras. In: Representation Theory o Finite Groups and Finite-Dimensional Algebras. Progress in Mathematics, vol. 95. Basel: Birkhäuser, 1991, Hovey M. Cotorsion pairs and model categories. In: Interactions Between Homotopy Theory and Algebra. Contemp Mathematics, vol Providence, RI: Amer Math Soc, 2007, Huang Z Y, Iyama O. Auslander-type conditions and cotorsion pairs. J Algebra, 2007, 318: Iyama O, Yoshino Y. Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent Math, 2008, 172: Krause H, Solberg. Applications o cotorsion pairs. J Lond Math Soc, 2003, 68: Orlov D. Triangulated categories o singularities and D-branes in Landau-Ginzburg models. Proc Steklov Inst Math, 2004, 246: Rickard J. Derived categories and stable equivalence. J Pure Appl Algebra, 1989, 61: Salce L. Cotorsion Theories or Abelian Categories. Cambridge: Cambridge University Press, Verdier J L. Catégories dérivées. Etat 0. In: Lecture Notes in Math, vol Berlin: Springer-Verlag, 1977,