Homological Aspects of the Dual Auslander Transpose II

Transcription

1 Homological Aspects of the Dual Auslander Transpose II Xi Tang College of Science, Guilin University of Technology, Guilin , Guangxi Province, P.R. China Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing , Jiangsu Province, P.R. China Abstract Let R and S be rings and R C S a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of -C-cotorsionfree modules and a subclass of the class of C-adstatic modules. Also we establish the relation between the relative homological dimensions of a module M and the corresponding standard homological dimensions of Hom(C, M). By investigating the properties of the Bass injective dimension of modules (resp. complexes), we get some equivalent characterizations of semi-tilting modules (resp. Gorenstein artin algebras). Finally we obtain a dual version of the Auslander-Bridger s approximation theorem. As a consequence, we get some equivalent characterizations of Auslander n-gorenstein artin algebras. Key Words: Semidualizing bimodules; -C-cotorsionfree modules; Bass classes; X -projective dimension; X -injective dimension; Bass injective dimension; (Strong) Ext-cograde, (Strong) Tor-cograde Mathematics Subject Classification: 16E10, 18G25, 16E05, 16E Introduction Semidualizing bimodules arise naturally in the investigation of various duality theories in commutative algebra. The study of such modules was initiated by Foxby in [18] and by Golod in [20]. Then Holm and White extended in [22] this notion to arbitrary associative rings, while Christensen in [11] and Kubik in [27] extended it to semidualizing complexes and quasidualizing modules respectively. The study of semidualizing bimodules or complexes was connected to the so-called Auslander classes and Bass classes defined by Avramov and Foxby in [5] and by Christensen in [11]. Semidualizing bimodules or complexes and the corresponding Auslander/Bass classes have been studied by many authors; see, for example, [5, 11, 12, 13, 14, 16, 22, 33] and so on. In order to dualize the important and useful notions of Auslander 1

2 transpose of modules and n-torsionfree modules, we introduced in [33] the notions of cotranspose of modules and n-cotorsionfree modules with respect to a semidualizing bimodule, and obtained several dual counterparts of interesting results. Based on these mentioned above, we study further homological properties of cotranspose of modules, n-cotorsionfree modules and related modules. The paper is organized as follows. In Section 2, we give some terminology and some preliminary results. In particular, we prove that if (R, m, k) is a commutative Gorenstein complete local ring with dim R > 0 and q is a prime ideal of R with height non-zero, then the tensor product of the injective envelopes of R/q and k is equal to zero. This gives a negative answer to an open question of Kubik posed in [27] about quasidualizing modules. Let R and S be rings and R C S a semidualizing bimodule. In Section 3, we prove that if the projective dimension of R C is finite, then the class of -C-cotorsionfree modules is contained in the right orthogonal class of R C; dually, if the projective dimension of C S is finite, then the above inclusion relation between these two classes of modules is reverse. Also we prove that there exists a Morita equivalence between the class of -C-cotorsionfree modules and a subclass of the class of C-adstatic modules. Finally, we establish the relation between the relative homological dimensions of a module M and the corresponding standard homological dimensions of Hom(C, M). In Section 4, we first give some criteria for computing the Bass injective dimension of modules in term of the vanishing of Ext-functors and some special approximations of modules. Then, motivated by the philosophy of [26], we introduce the notion of semi-tilting bimodules in general case, and prove that R C S is right semi-tilting if and only if the Bass injective dimension of R R is finite. In Section 5, we extend the Bass class and the Bass injective dimension of modules with respect to C to that of homologically bounded complexes. We show that a homologically bounded complex has finite Bass injective dimension if and only if it admits a special quasi-isomorphism in the derived category of the category of modules. As an application of this result, we get some equivalent characterizations of Gorenstein artin algebras. In Section 6, we first introduce the notions of (strong) Ext-cograde and Torcograde of modules with respect to C. Then we obtain a dual version of the Auslander-Bridger s approximation theorem ([17, Proposition 3.8]) as follows. For any left R-module M and n 1, if the Tor-cograde of Ext i R(C, M) with respect to C is at least i for any 1 i n, then there exists a left R-module U and a homomorphism f : U M of left R-modules satisfying the following properties: 2

3 (1) The injective dimension of U relative to the class of C-projective modules is at most n, and (2) Ext i R(C, f) is bijective for any 1 i n. As an application of this result, we prove that for any n 1, the strong Ext-cograde of Tor S i (C, N) with respect to C is at least i for any left S-module N and 1 i n if and only if the strong Tor-cograde of Ext i R(C, M) with respect to C is at least i for any left R-module M and 1 i n. Furthermore, we get some equivalent characterizations of Auslander n-gorenstein artin algebras. 2. Preliminaries Throughout this paper, R and S are fixed associative rings with unites. We use Mod R (resp. Mod S op ) to denote the class of left R-modules (resp. right S-modules). Let M Mod R. We use pd R M, fd R M and id R M to denote the projective, flat and injective dimensions of M respectively, and use Add R M (resp. Prod R M) to denote the subclass of Mod R consisting of all direct summands of direct sums (resp. direct products) of copies of M. We use 0 M I 0 (M) f 0 I 1 (M) f 1 f i 1 I i (M) f i (1.1) to denote a minimal injective resolution of M. For any n 1, coω n (M) := Im f n 1 is called the n-th cosyzygy of M, and in particular, coω 0 (M) := M. Definition 2.1. ([22]). (1) An (R-S)-bimodule R C S is called semidualizing if the following conditions are satisfied: (a1) R C admits a degreewise finite R-projective resolution. (a2) C S admits a degreewise finite S-projective resolution. Rγ (b1) The homothety map R R R Hom S op(c, C) is an isomorphism. γ (b2) The homothety map S S S S HomR (C, C) is an isomorphism. (c1) Ext 1 R (C, C) = 0. (c2) Ext 1 Sop(C, C) = 0. (2) A semidualizing bimodule R C S is called faithful if the following conditions are satisfied: (f1) If M Mod R and Hom R (C, M) = 0, then M = 0. (f2) If N Mod S op and Hom S op(c, N) = 0, then N = 0. Typical examples of semidualizing bimodules include the free module of rank one, dualizing modules over a Cohen-Macaulay local ring and the ordinary Matlis dual bimodule Λ D(Λ) Λ of Λ Λ Λ over an artin algebra Λ. Any semidualizing bimodule over commutative rings is faithful ([22, Proposition 3.1]). Semidualizing bimodules occur in the literature with several different names, e.g., in the work of [18, 20, 29, 34]. 3

4 Let R be a commutative noetherian local ring with maximal ideal m and residue field k = R/m. According to [27], an artinian R-module T is called quasidualizing if the homothety ˆR Hom R (T, T ) is an isomorphism (where ˆR is the m-adic completion of R) and Ext i1 R (T, T ) = 0. It was proved in [27, Lemma 3.11] that if L and T are R-modules with T quasidualizing such that Hom R (T, L) = 0, then L = 0. Motivated by this result and [22, Lemma 3.1], an open question was posed in [27] as follows. Question 2.2. ([27, Question 3.12]) Let R be a commutative noetherian local ring. If L and T are R-modules with T quasidualizing such that T R L = 0, then L = 0? The following result shows that the answer to this question is negative in general. Proposition 2.3. Let R be a commutative noetherian complete local ring with maximal ideal m and residue field k = R/m. If R is Gorenstein (that is, id R R < ) with dim R > 0, then E 0 (R/q) R E 0 (k) = 0 for any prime ideal q with ht(q) > 0, where ht(q) is the height of q. Proof. By [30, Theorem 4.2], E 0 (k) is quasidualizing. Since R is Gorenstein, it follows from [7, Fundamental Theorem] that E i (R) = ht(p)=i E(R/p) with p Spec(R) (the prime spectrum of R) for any i 0. In particular, E 0 (R) = ht(p)=0 E(R/p) with p Spec(R). On the other hand, for any p, q Spec(R) with ht(p) = 0 and ht(q) > 0, we have Hom R (E 0 (R/q), E 0 (R/p)) = 0. So Hom R (E 0 (R/q), E 0 (R)) = 0 and Hom R (E 0 (R/q), R) = 0. Thus we have Hom R (E 0 (R/q) R E 0 (k), E 0 (k)) = HomR (E 0 (R/q), Hom R (E 0 (k), E 0 (k))) (by the adjoint isomorphism theorem) = HomR (E 0 (R/q), R) (by [30, Theorem 4.2]) = 0. Because E 0 (k) is an injective cogenerator for Mod R, E 0 (R/q) R E 0 (k) = 0. From now on, R C S is a semidualizing bimodule. For convenience, we write ( ) = Hom R (C, ) and R C = {M Mod R Ext i1 R (C, M) = 0}. Let M Mod R and N Mod S. Then we have the following two canonical valuation homomorphisms: θ M : C S M M defined by θ M (c f) = f(c) for any c C and f M ; and µ N : N (C S N) 4

5 defined by µ N (x)(c) = c x for any x N and c C. Following [36], M (resp. N) is called C-static (resp. C-adstatic) if θ M (resp. µ N ) is an isomorphism. We denote by Stat(C) and Adst(C) the class of all C-static modules and the class of all C-adstatic modules, respectively. Definition 2.4. ([22]). The Bass class B C (R) with respect to C consists of all left R-modules M satisfying (B1) M R C, (B2) Tor S 1(C, M ) = 0, and (B3) M Stat(C), that is, θ M is an isomorphism in Mod R. The Auslander class A C (S) with respect to C consists of all left S-modules N satisfying (A1) Tor S i1(c, N) = 0, (A2) C S N R C, and (A3) N Adst(C), that is, µ N is an isomorphism in Mod S. Definition 2.5. ([33]). Let M Mod R and n 1. (1) ctr C M := Coker f 0 is called the cotranspose of M with respect to R C S, where f 0 is as in (1.1). (2) M is called n-c-cotorsionfree if Tor S 1in(C, ctr C M) = 0; and M is called -C-cotorsionfree if it is n-c-cotorsionfree for all n. The class of all -Ccotorsionfree modules is denoted by ct (R). In particular, every module in Mod R is 0-C-cotorsionfree. By [33, Proposition 3.2], a module is 2-C-cotorsionfree if and only if it is C-static. Let W X be subclasses of Mod R. Recall from [2] that W is called a generator for X if for any X X, there exists an exact sequence 0 X W X 0 in Mod R with W W and X X ; W is called an Ext-projective generator for X if W is a generator for X and Ext i1 R (W, X) = 0 for any X X and W W. Also recall that X is called coresolving if it is closed under extensions, cokernels of monomorphisms and it contains all injective modules in Mod R. Let M Mod R. An exact sequence (of finite or infinite length): X n X 1 X 0 M 0 in Mod R is called an X -resolution of M if all X i are in X ; furthermore, such an X - resolution is called proper if it remains exact after applying the functor Hom R (X, ) for any X X. The X -projective dimension X -pd R M of M is defined as inf{n there exists an X -resolution 0 X n X 1 X 0 M 0 of M in 5

6 Mod R}. Dually, the notions of an X -coresolution, an X -coproper coresolution and the X -injective dimension X -id R M of M are defined. Definition 2.6. ([15]) A module M Mod R is called Gorenstein projective if there exists an exact sequence of projective modules: P := P 1 P 0 P 0 P 1 in Mod R satisfying the conditions: (1) it remains exact after applying the functor Hom R (, P ) for any projective module P in Mod R; and (2) M = Im(P 0 P 0 ). Dually, the notion of Gorenstein injective modules is defined. We use GP(R) (resp. GI(R)) to denote the subclass of Mod R consisting of Gorenstein projective (resp. Gorenstein injective) modules. Fact 2.7. (1) B C (R) is coresolving and Add R C is an Ext-projective generator for B C (R) (see [22, Proposition 5.1(b),Theorem 6.2] and [33, Proposition 3.7]). (2) When C is a dualizing module over a local Cohen-Macaulay ring R, B C (R) actually is exactly the class of modules admitting finite Gorenstein injective dimensions (see [16, Corollary 2.6]). However, the following example illustrates that these two classes of modules are different in general. Example 2.8. Let Λ be the finite-dimensional algebra over a field defined by the following quiver and relation: (rad Λ) 2 = 0. Take Λ C = and M = Then by [35, Example 3.1] and [33, Theorem 3.9], Λ C End(Λ C) is a semidualizing bimodule and M B C (Λ). But an easy computation shows that the Gorenstein injective dimension of M is infinite. Let E be a subcategory of an abelian category A. Recall from [15] that a sequence: 4 S : S 1 S 2 S 3 in A is called Hom A (E, )-exact (resp. Hom A (, E)-exact) if Hom A (E, S) (resp. Hom A (S, E)) is exact for any object E in E. An epimorphism (resp. a monomorphism) in A is called E-proper (resp. E-coproper) if it is Hom A (E, )-exact (resp. Hom A (, E)-exact). Definition 2.9. ([24]) Let E and T be subcategories of an abelian category A. Then T is called E-coresolving in A if the following conditions are satisfied. 6 5,

7 (1) T admits an E-coproper cogenerator C, that is, C T, and for any object T in T, there exists a Hom A (, E)-exact exact sequence 0 T C T 0 in A such that C is an object in C and T is an object in T. (2) T is closed under E-coproper extensions, that is, for any Hom A (, E)-exact exact sequence 0 A 1 A 2 A 3 0 in A, if both A 1 and A 3 are objects in T, then A 2 is also an object in T. (3) T is closed under cokernels of E-coproper monomorphisms, that is, for any Hom A (, E)-exact exact sequence 0 A 1 A 2 A 3 0 in A, if both A 1 and A 2 are objects in T, then A 3 is also an object in T. Dually, the notions of E-proper generators and E-resolving subcategories are defined. 3. Relative Homological Dimensions Holm and White obtained in [22] some equivalent characterizations of B C (R) in terms of the so-called C-projective and C-flat modules. Similar results were also proved by Enochs and Holm in [14]. Recently, we proved in [33, Theorem 3.9] that B C (R) = ct (R) R C. In the beginning of this section, we investigate the further relation among ct (R), R C and B C (R). Proposition 3.1. (1) If pd R C <, then ct (R) R C. (2) If pd S op C <, then R C ct (R). Proof. (1) Let M ct (R). Then by [33, Proposition 3.7], there exists an exact sequence: W n W n 1 W 0 M 0 in Mod R with all W i Add R C. Put M i = Im(W i W i 1 ) for any i 1. We may assume pd R C = n < by assumption. Since W i R C by [33, Lemma 2.5(1)], Ext i R(C, M) = Ext i+n R (C, M n) = 0 for any i 1 and M R C. (2) Let M R C and pd S op C = n <. Then we get an exact sequence: 0 coω i (M) I i (M) coω i+1 (M) 0 in Mod S for any i 0. Note that fd S op C = pd S op C = n because C is finitely presented as a right S-module. Since Tor S i1(c, I ) = 0 for any injective left R- module I by [33, Lemma 2.5(2)], Tor S j (C, coω i (M) ) = Tor S j+n(c, coω i+n (M) ) = 0 for any i 0 and j 1; in particular, Tor S 1 (C, coω 2 (M) ) = 0. Then we have the 7

8 following diagram with exact rows: 0 C S coω 1 (M) C S I 1 (M) θ coω 1 (M) θ I 1 (M) 0 coω 1 (M) I 1 (M). Because θ I 1 (M) is an isomorphism by [33, Lemma 2.5(2)], θ coω 1 (M) is a monomorphism. So coω 1 (M) is 2-C-cotorsionfree by [33, Lemma 4.1(1)]. On the other hand, because Tor S 1 (C, coω 1 (M) ) = 0 by the above argument, we have the following commutative diagram with exact rows: 0 C S M C S I 0 (M) C S coω 1 (M) 0 θ M θ I 0 (M) θ coω 1 (M) 0 M I 0 (M) coω 1 (M) 0. Because θ I 0 (M) is an isomorphism by [33, Lemma 2.5(2)], applying the snake lemma we have that θ M is also an isomorphism and M is 2-C-cotorsionfree. So by [33, Corollary 3.8], there exists an exact sequence 0 M 1 W 0 M 0 in Mod R with W 0 Add R C and Ext 1 R(C, M 1 ) = 0. Thus M 1 R C since M R C. Then by a argument similar to the above, we get an exact sequence 0 M 2 W 1 M 1 0 in Mod R with W 1 Add R C and M 2 R C. Continuing this procedure, we get a proper Add R C-resolution: W n W n 1 W 0 M 0 of M in Mod R. Thus M ct (R) by [33, Proposition 3.7]. The following result extends [35, Corollary 2.16]. Corollary 3.2. (1) If pd R C <, then B C (R) = ct (R). (2) If pd S op C <, then B C (R) = R C. Proof. It is an immediate consequence of Proposition 3.1 and [33, Theorem 3.9]. We write Ker Ext i1 S (, C+ ) = {N Mod S Ext i1 S (N, C+ ) = 0} and H(C) = Adst(C) Ker Ext i1 S (, C+ ), where ( ) + = Hom Z (, Q/Z) with Z the additive group of integers and Q the additive group of rational numbers. In the following result, we provide a viewpoint from Morita equivalence for ct (R). 8

9 Theorem 3.3. There exists an equivalence of categories: ct (R) ( ) C S H(C). Proof. According to [36, 2.4], the functors ( ) and C S induce an equivalence between the category of all 2-C-cotorsionfree modules and Adst(C). So it suffices to show that ( ) (resp. C S ) maps ct (R) (resp. H(C)) to H(C) (resp. ct (R)). Let M ct (R). Then by [36, 2.4], we have M Adst(C). By [33, Proposition 3.7] there exists a proper Add R C-resolution: W n W n 1 W 0 M 0 (3.1) of M in Mod R. Thus we get an exact sequence: W n W n 1 W 0 M 0 in Mod S. Applying C S to this exact sequence gives back the sequence (3.1). Then we get easily that Tor S i1(c, M ) = 0 because Tor S i1(c, W j ) = 0 for any j 0 by [33, Lemma 2.5(2)]. It follows from the mixed isomorphism theorem that Ext i1 S (M, C + ) = [Tor S i1(c, M )] + = 0. So M Ker Ext i1 S (, C+ ) and M H(C). Conversely, let N H(C). Then (C S N) = N. It follows from the mixed isomorphism theorem that [Tor S i1(c, (C S N) )] + = [Tor S i1(c, N)] + = Ext i1 S (N, C+ ) = 0 and Tor S i1(c, (C S N) ) = 0. In addition, C S N is 2-C-cotorsionfree by [36, 2.4]. Thus we conclude that C S N is -C-cotorsionfree by [33, Corollary 3.4]. Following [22], set F C (R) = {C S F F is flat in Mod S}, P C (R) = {C S P P is projective in Mod S}, I C (S) = {Hom R (C, I) I is injective in Mod R}. The modules in F C (R), P C (R) and I C (S) are called C-flat, C-projective and C- injective respectively. For a module M Mod R, we use lim M (R) to denote the subcategory of Mod R consisting of all modules isomorphic to direct summands of a direct limit of a family modules in which each is a finite direct sum of copies of M. Proposition 3.4. (1) F C (R) = lim C (R) (2) P C (R) = Add R C. (3) I C (S) = Prod S E with R E an injective cogenerator for Mod R. 9

10 Proof. (1) It is well known that a module in Mod S is flat if and only if it is in lim S (S). Because the functor C S commutes with direct limits, we get easily F C (R) lim C (R). Now let M lim C (R). Then M B C (R) by [22, Proposition 4.2(a)]. Because R C admits a degreewise finite R-projective resolution, Hom R (C, ) commutes with direct limits. So Hom R (C, M) is in lim S (S), that is, Hom R (C, M) is a flat left S-module. Then by [22, Lemma 5.1(a)], we have M F C (R), and thus lim C (R) F C (R). (2) and (3) See [28, Proposition 2.4]. The following result establishes the relation between the relative homological dimensions of a module M and the corresponding standard homological dimensions of M. It extends [31, Theorem 2.11]. Theorem 3.5. (1) fd S M F C (R)-pd R M for any M Mod R, the equality holds if M ct (R). (2) pd S M P C (R)-pd R M for any M Mod R, the equality holds if M ct (R). (3) id R C S N I C (S)-id S N for any N Mod S, the equality holds if N A C (S). Proof. (1) Let M Mod R with F C (R)-pd R M = n <. Then there exists an exact sequence: 0 L n L 1 L 0 M 0 (3.2) in Mod R with all L i in lim C (R) by Proposition 3.4(1). Because R C admits a degreewise finite R-projective resolution, Ext i R(C, ) commutes with direct limits for any i 0. Also notice that ( R C) = S and C R C, so we have that L i is in lim S (S) (that is, L i is left S-flat) and L i R C for any 0 i n. Applying the functor Hom R (C, ) to the exact sequence (3.2) we obtain the following exact sequence: 0 L n L 1 L 0 M 0 in Mod S, and so fd S M n. (2) Let M Mod R with P C (R)-pd R M = n <. Then there exists an exact sequence: 0 C n C 1 C 0 M 0 (3.3) in Mod R with all C i Add R C by Proposition 3.4(2). Because all C i are projective left S-modules and Add R C R C by [33, Lemma 2.5(1)], applying the functor ( ) 10

11 to the exact sequence (3.3) we get the following exact sequence: 0 C n C 1 C 0 M 0 in Mod S, and so pd S M n. Now suppose M ct (R). Then C S M = M. By [33, Corollary 3.4(3)], we have Tor S i1(c, M ) = 0. We will prove the equalities in (1) and (2) hold. (1) Assume fd S M = n <. Then there exists an exact sequence: 0 F n F 1 F 0 M 0 in Mod S with all F i flat. Applying the functor C S to it we get an exact sequence: 0 C S F n C S F 1 C S F 0 C S M ( = M) 0 in Mod R with all C S F i in F C (R), so we have F C (R)-pd R M n. (2) Assume pd S M = n <. Then there exists an exact sequence: 0 P n P 1 P 0 M 0 in Mod S with all P i projective. Applying the functor C S to it we get an exact sequence: 0 C S P n C S P 1 C S P 0 C S M ( = M) 0 in Mod R with all C S P i in P C (R), and so P C (R)-pd R M n. (3) Let N Mod S with I C (S)-id S N = n < and R E be an injective cogenerator for Mod R. Then there exists an exact sequence: 0 N I 0 I 1 I n 0 (3.4) in Mod S with all I i in Prod S E by Proposition 3.4(3). Because C S admits a degreewise finite S-projective resolution, Tor S j (C, ) commutes with direct products for any j 0. Then by [33, Lemma 2.5(2)], C S I i ( Prod R E) is injective in Mod R and Tor S j1(c, I i ) = 0 for any 0 i n. Applying the functor C S to the exact sequence (3.4) we obtain the following exact sequence: 0 C S N C S I 0 C S I 1 C S I n 0 in Mod R, and so id R C S N n. Now suppose N A C (S). Then N = (C S N) and C S N R C. If id R C S N = n <, then there exists an exact sequence: 0 C S N E 0 E 1 E n 0 11

12 in Mod R with all E i injective. Applying the functor Hom R (C, ) to it we get an exact sequence: 0 (C S N) ( = N) E 0 E 1 E n 0 in Mod S with all E i I C (S), and so I C (S)-id S N n. For a subclass X of Mod R, we write id R X := sup{id R X X X }. application of Theorem 3.5, we get the following As an Proposition 3.6. (1) sup{f C (R)-pd R M M ct with F C (R)-pd R M < } id R F C (R). (2) sup{p C (R)-pd R M M ct with P C (R)-pd R M < } id R P C (R). Proof. (1) Let id R F C (R) = n < and M ct (R) with F C (R)-pd R M = m <. By Theorem 3.5(1), fd S M = m and there exists an exact sequence: 0 F m Q m 1 Q 1 Q 0 M 0 (3.5) in Mod S with F m flat and all Q i projective. Because C S M = M and Tor S j1(c, M ) = 0 by [33, Corollary 3.4(3)], applying the functor C S to the exact sequence (3.5), we get the following exact sequence: 0 C S F m C S Q m 1 C S Q 1 C S Q 0 C S M ( = M) 0 (3.6) in Mod R with C S F m in F C (R) (= lim C (R) by Proposition 3.4(1)) and all C S Q i in P C (R) (= Add R C by Proposition 3.4(2)). Notice that R C admits a degreewise finite R-projective resolution and C R C, so Ext j1 R (C S Q i, C S F m ) = 0 for any 0 i m 1. Suppose m > n. Because id R C S F m n, it follows from the exact sequence (3.6) that Ext 1 R(K, C S F m ) = Ext m R (M, C S F m ) = 0, where K = Coker(C S F m C S Q m 1 ). Thus the exact sequence 0 C S F m C S Q m 1 K 0 splits and K P C (R)( F C (R)). It induces that F C (R)-pd R M m 1, which is a contradiction. Thus we conclude that m n. (2) It is similar to the proof of (1), so we omit it. Note that R R R is a semidualizing bimodule. Let R be a left noetherian ring and RC S = R R R. Then we have the following facts: (1) F C (R) and P C (R) are the subclasses of Mod R consisting of flat modules and projective modules respectively, and F C (R)-pd R M = fd R M and P C (R)- pd R M = pd R M for any M Mod R. (2) id R F C (R) = id R R and id R P C (R) = id R R by [6, Theorem 1.1]. (3) ct = Mod R by [33, Proposition 3.7]. 12

13 So by Proposition 3.6, we immediately have the following Corollary 3.7. For a left noetherian ring R, we have (1) sup{fd R M M Mod R with fd R M < } id R R. (2) ([6, Proposition 4.3]) sup{pd R M M Mod R with pd R M < } id R R. In the rest of this section, for a module M Mod R, in case P C (R)-pd R M <, we establish the relation between P C (R)-pd R M and some standard homological dimensions of related modules. Lemma 3.8. If M ct (R) and N R C, then for any i 0, we have an isomorphism of abelian groups: Ext i R(M, N) = Ext i S(M, N ). Proof. We proceed by induction on i. Let i = 0. Since M ct (R), C S M = M. It follows from the adjoint isomorphism theorem that Hom R (M, N) = Hom R (C S M, N) = Hom S (M, N ). Indeed the isomorphism is natural in M and N. Now suppose i 1. The induction hypothesis implies that there exists a natural isomorphism: Ext j R (L, H) = Ext j S (L, H ) for any L ct (R), H R C and 0 j i 1. Because N R C by assumption, coω 1 (N) R C and we have an exact sequence: 0 N I 0 (N) coω 1 (N) 0. Applying the functor Hom S (M, ) to it yields a commutative diagram with exact rows: Ext i 1 R (M, I0 (N)) Ext i 1 R (M, coω1 (N)) Ext i R(M, N) 0 Ext i 1 S (M, I 0 (N) ) Ext i 1 S (M, coω 1 (N) ) Ext i S(M, N ) Ext i S(M, I 0 (N) ). By the induction hypothesis, the first two columns in the above diagram are natural isomorphisms. Since M ct (R) by assumption, by the mixed isomorphism theorem and [33, Corollary 3.4(3)] we have Ext i S(M, I 0 (N) ) = Hom R (Tor i S(C, M ), I 0 (N)) = 0. It follows that Ext i R(M, N) = Ext i S(M, N ) naturally. We also need the following criterion. Lemma 3.9. Let M Mod R admit a degreewise finite R-projective resolution. If P C (R)-pd R M <, then P C (R)-pd R M = sup{i 0 Ext i R(M, C) 0}. 13

14 Proof. Let P C (R)-pd R M = n < and 0 C n C 1 C 0 M 0 be an exact sequence in Mod R with all C i in P C (R)(= Add R C). It is easy to see that Ext i R(M, C) = 0 for i n + 1. Put M n 1 = Coker(C n C n 1 ). If Ext n R(M, C) = 0, then by [19, Lemma 3.1.6], we have that Ext n R(M, C i ) = 0 and Ext 1 R (C j, C i ) = 0 for any 0 i, j n. So Ext 1 R(M n 1, C n ) = Ext n R(M, C n ) = 0 and the exact sequence: 0 C n C n 1 M n 1 0 splits. It implies that M n 1 P C (R) and P C (R)-pd R M n 1, which is a contradiction. So we conclude that Ext n R(M, C) 0. Now we are in a position to give the following Proposition Let M Mod R admit a degreewise finite R-projective resolution. If P C (R)-pd R M <, then P C (R)-pd R M min{id R C, id S S, pd R M, pd S M }. Proof. Let M Mod R with P C (R)-pd R M <. Then M ct (R) by [33, Proposition 3.7]. So Ext i R(M, C) = Ext i S(M, C ) = Ext i S(M, S) for any i 0 by Lemma 3.8, and hence sup{i 0 Ext i R(M, C) 0} min{id R C, id S S, pd R M, pd S M }. Now the assertion follows from Lemma 3.9. The following example shows that the finiteness of P C (R)-pd R M is necessary for the conclusion of Proposition Example Let G be a finite group and k a field such that the characteristic of k divides G. Take R = S = C = kg. By [4, Theorem 3.3 and Propositon 3.10], the group algebra kg is a non-semisimple symmetric artin algebra. Then id R C = 0 and there exists a kg-module M with P C (R)-pd R M infinite. 4. The Bass Injective Dimension of Modules For a module M in Mod R, we study in this section the properties of the Bass injective dimension B C (R)-id R M of M. We begin with the following easy observation. Lemma 4.1. For any M Mod R, if B C (R)-id R M < and M R C, then M B C (R). Proof. It is easy to get the assertion by using induction on B C (R)-id R M. Now we give some criteria for computing B C (R)-id R M in terms of the vanishing of Ext-functors and some special approximations of M. 14

15 Theorem 4.2. Let M Mod R with B C (R)-id R M < and n 0. Then the following statements are equivalent. (1) B C (R)-id R M n. (2) coω m (M) B C (R) for m n. (3) Ext n+1 R (C, M) = 0. (4) There exists an exact sequence: 0 M X M W M 0 in Mod R such that X M B C (R) and P C (R)-id R W M n 1. (5) There exists an exact sequence: 0 X M W M M 0 in Mod R such that X M B C (R) and P C (R)-id R W M n. Proof. (1) (2) follows from [22, Theorem 6.2] and [24, Theorem 4.8], (2) (3) follows from the dimension shifting, and (4) (1) follows from the fact that P C (R) B C (R). (3) (1) Let M Mod R with B C (R)-id R M <. Then B C (R)-id R coω n (M) < by [22, Theorem 6.2] and [24, Theorem 4.8]. If Ext n+1 R (C, M) = 0, then coω n (M) R C, and so coω n (M) B C (R) by Lemma 4.1. It follows that B C (R)- id R M n. (1) (4) By [22, Theorem 6.2], B C (R) is closed under extensions. By [33, Proposition 3.7], it is easy to see that P C (R)(= Add R C) is a P C (R)-proper generator for B C (R). Then the assertion follows from [24, Theorem 3.7]. (4) (5) Assume that exists an exact sequence: 0 M X M W M 0 in Mod R such that X M B C (R) and P C (R)-id R W M n 1. By [33, Proposition 3.7], there exists an exact sequence: 0 X W 0 X M 0 15

16 in Mod R with W 0 P C (R) and X B C (R). Now consider the following pull-back diagram: 0 0 X X 0 W M W 0 W M 0 0 M X M W M Then the leftmost column in the above diagram is the desired sequence. (5) (4) Assume that there exists an exact sequence: 0 X M W M M 0 in Mod R such that X M B C (R) and P C (R)-id R W M n. Then there exists an exact sequence: 0 W M W 0 W 0 in Mod R with W 0 P C (R) and P C (R)-id R W push-out diagram: 0 n 1. Consider the following 0 0 X M W M M 0 0 X M W 0 X 0 W W 0 0. It follows from [22, Theorem 6.2] and the exactness of middle row in the above diagram that X B C (R). So the rightmost column in the above diagram is the desired sequence. 16

17 Remark 4.3. The only place where the assumption that B C (R)-id R M < in Theorem 4.2 is used is in showing (3) (1). If the given semidualizing module R C S is faithful, then a module in Mod R with finite Bass injective dimension is in B C (R) by [22, Theorem 6.3]. However, this property does not hold true in general. Example 4.4. Let Λ be a finite-dimensional algebra over an algebraically closed field given by the quiver: 1 2 Put C = I(1) I(2). Then Λ C Λ is a semidualizing bimodule, but non-faithful since Hom Λ (C, S(2)) = 0. We have an exact sequence 0 S(2) I(2) I(1) 0 in Mod Λ. Both I(1) and I(2) are obviously in B C (Λ). But S(2) is not in B C (Λ) because S(2) is not 2-C-cotorsionfree. Motivated by [26, Definition 2.4 and Lemma 2.5], we introduce the following Definition 4.5. A semidualizing bimodule R C S is called left (resp. right) semitilting if pd R C < (resp. pd S op C < ). In the following, we will give an equivalent characterization of right semi-tilting bimodules in terms of the finiteness of the Bass injective dimension of R R. We need the following two lemmas. Let C be the subclass of Mod R consisting of direct sums of copies of C. Lemma 4.6. Let M Mod R with P C (R)-id R M n(< ). If K Mod R is isomorphic to a direct summand of M, then P C (R)-id R M n. Proof. Note that P C (R) = Add R C by Proposition 3.4(2). It is clear that P C (R) P C (R). In addition, it is not difficult to verify that P C (R) is P C (R)-coresolving in Mod R with P C (R) a P C (R)-coproper cogenerator in the sense of [24]. Now the assertion follows from [24, Corollary 4.9]. We use add R C to denote the subclass of Mod R consisting of direct summands of finite direct sums of copies of C. Lemma 4.7. Let M Mod R be finitely generated and n 0. If P C (R)-id R M n, then there exists an exact sequence: in Mod R with all C i in add R C. 0 M C 0 C 1 C n 0 17

18 Proof. Let P C (R)-id R M n and 0 M α0 D 0 α 1 D 1 α 2 α n D n 0 (4.1) be an exact sequence in Mod R with all D i in Add R C (= P C (R)). Put K i = Im α i for any 0 i n. There exists a module G 0 Add R C such that D 0 G 0 is a direct sum of copies of C, so we get a Hom R (, C)-exact exact sequence: 0 M β0 D 0 G 0 β 1 D 1 G 0 β 2 D 2 α 3 where β 0 = ( ( ) ) α 0 0, β 1 α = G 0 α n D n 0, and β 2 = (α 2, 0). Then Im β 1 = K 1 G 0 and Im β 2 = K 2. Because M is finitely generated by assumption, there exist C 0 add R C and H 0 Add R C such that D 0 G 0 = C 0 H 0 and Im α 0 C 0. So we get an exact sequence: 0 M C 0 L 0 0 (4.2) in Mod R with L 0 H 0 = Im β 1. Consider the following push-out diagram with the middle row Hom R (, C)-exact exact and the leftmost column splitting: 0 H 0 0 Im β 1 0 L 0 0 H 0 D 1 G 0 X 1 K 2 0 K Then the middle column in the above diagram is Hom R (, C)-exact exact. From the proof of Lemma 4.6, we know that Add R C (= P C (R)) is C-coresolving in Mod R. So X 1 Add R C. Combining the exact sequences (4.1), (4.2) with the bottom row in the above diagram, we get an exact sequence: 0 M C 0 X 1 D 2 α 3 α n D n 0 in Mod R with C 0 add R C and X 1 Add R C. Repeating the above argument with Im(C 0 X 1 ) replacing M, we get an exact sequence: 0 M C 0 C 1 X 2 D 3 α 4 α n D n 0 18

19 in Mod R with C 0, C 1 add R C and X 2 Add R C. Continuing this procedure, we finally get an exact sequence: 0 M C 0 C 1 C n 0 in Mod R with all C i in add R C. We are now in a position to prove the following Theorem 4.8. (1) If R C S is right semi-tilting, then B C (R) = R C. (2) If S is a left coherent ring, then R C S is right semi-tilting with pd S op C n if and only if B C (R)-id R R n. Proof. (1) It follows from Corollary 3.2 and [33, Theorem 3.9]. (2) It is easy to see that B C (R)-id R R P C (R)-id R R = pd S op C. Now the necessity is clear. Conversely, if B C (R)-id R R = n <, then by Theorem 4.2, there exists a split exact sequence: 0 X W R 0 in Mod R such that X B C (R) and P C (R)-id R W n. So W = X R and P C (R)- id R R n by Lemma 4.6. It follows from Lemma 4.7 that there exists an exact sequence: 0 R C 0 C 1 C n 0 in Mod R with all C i in add R C. Applying the functor Hom R (, C) to it we get the following exact sequence: 0 Hom R (C n, C) Hom R (C 1, C) Hom R (C 0, C) C 0 in Mod S op with all Hom R (C i, C) projective. So R C S is right semi-tilting with pd S op C n. Compare the following result with Lemma 3.9. Corollary 4.9. If R C S is left and right semi-tilting, then for every M Mod R, B C (R)-id R M = sup{i 0 Ext i R(C, M) 0} <. Proof. Let R C S be left and right semi-tilting. Then pd R C < and pd S op C <. Put sup{i 0 Ext i R(C, M) 0} = n. Then n <. It is easy to see that RC -id R M n. So B C (R)-id R M n by Theorem 4.8(1). We will use induction on n to prove B C (R)-id R M n. If n = 0, then M R C. It follows from Theorem 4.8(1) that M B C (R). Now suppose n 1. Then sup{i 0 Ext i R(C, coω 1 (M)) 0} = n 1. So B C (R)-id R coω 1 (M) = n 1 by the induction hypothesis, and hence B C (R)-id R M n. 19

20 5. The Bass Injective Dimension of Complexes In this section, we extend the Bass injective dimension of modules to that of complexes in derived categories. A cochain complex M is a sequence of modules and morphisms in Mod R of the form: M n 1 d n 1 M n d n M n+1 such that d n d n 1 = 0 for any n Z, and the shifted complex M [m] is the complex with M [m] n = M m+n and d n M [m] = ( 1)m d n m+n. Any M Mod R can be considered as a complex having M in its 0-th spot and 0 in its other spots. We use C(R) and D b (R) to denote the category of cochain complexes and the derived category of complexes with bounded finite homologies of Mod R, respectively. According to [10, Appendix], the supremum, the infimum and the amplitude of a complex M are defined as follows: sup M = sup{n Z H n (M ) 0}, inf M = inf{n Z H n (M ) 0}, amp M = sup M inf M. The Auslander category with respect to a dualizing complex was defined in [12]. Dually we define the Bass class of complexes with respect to C as follows. Definition 5.1. A full subcategory B C (R) of Db (R) consisting of complexes M is called the Bass class with respect to C if the following conditions are satisfied: (1) R Hom R (C, M ) D b (R). (2) C L S R Hom R(C, M ) M is an isomorphism in D b (R). Let M C(R) and n Z. The hard left-truncation n M of M at n is given by: n M := 0 0 M n d n M n+1 d n+1 M n+2. Let M D b (R) with H(M ) 0 and inf M = i. Taking an injective resolution I of M. We define the injective complex vi = ( i+1 I )[1], which is unique up to an injective summand in degree i. In general, we have that H t (vi ) = H t (I [1]) if t i + 1. In particular, when M is a module M, vi is isomorphic to coω 1 (M) in D b (R). Remark 5.2. (1) Let M D b (R). We see from the definition of vi that there exists a distinguished triangle in D b (R) of the form: vi [ 1] M I i [ i] vi. 20

21 (2) It is routine to check that B C (R) forms a triangulated subcategory of Db (R). Thus for an injective complex I, I B C (R) if and only if vi B C (R). Lemma 5.3. Let M Mod R. Then the following statements are equivalent. (1) B C (R)-id R M <. (2) M B C (R). Proof. (1) (2) Let B C (R)-id R M < and 0 M Y 0 Y 1 Y n 0 be an exact sequence in Mod R with all Y i in B C (R). Then by Remark 5.2(2) and [21, p.41, Corollary 7.22], we have M B C (R). (2) (1) Let M B C (R) and I be an injective resolution of M. Then I B C (R) and R Hom R(C, M) D b (R). Put s = sup R Hom R (C, M). Because H i (R Hom R (C, v s I )) = H i+s (R Hom R (C, I )) = 0 for any i 1. It implies that coω s (M) R C. By Remark 5.2(2) we have that v s I = coω s (M) and v s I B C (R), so coωs (M) B C (R), and hence C L S R Hom R(C, coω s (M)) coω s (M) is an isomorphism in D b (R), equivalently C S coω s (M) = coω s (M) and Tor S i1(c, coω s (M) ) = 0. It follows that coω s (M) B C (R) and B C (R)- id R M s. We define the Bass injective dimension of complexes in D b (R) as follows. Definition 5.4. Let M be a complex in D b (R). We define the Bass injective dimension of M as { B C(R)- id M := sup R Hom R (C, M ) if M B C (R), + if M / B C (R). In the following result, we give an equivalent characterization when the Bass injective dimension of complexes is finite. Theorem 5.5. Let M be a complex in D b (R). Then the following statements are equivalent. (1) B C (R)-id M <. (2) There exists an isomorphism M Y in D b (R) with Y a bounded complex consisting of modules in B C (R). Proof. (2) (1) The assertion follows from the fact that a complex Y of finite length consisting of modules in B C (R) is in B C (R). (1) (2) Let B C (R)-id M <. Then M B C (R). We will proceed by induction on amp M. If amp M = 0, then there exists T Mod R such that 21

22 M = T [ s], where s = sup M. Since B C (R)-id R T < by Lemma 5.3, we have a quasi-isomorphism T Y with Y := 0 Y 0 Y 1 Y n 0 a bounded complex and all Y i in B C (R). Then the complex Y [ s] is the desired complex. Now suppose amp M 1. By Remark 5.2(1), there exists a distinguished triangle: vi [ 1] M I i α [ i] vi in D b (R). Since amp vi < amp M, by the induction hypothesis, there exists an isomorphism β : vi Y 1 in D b (R) with Y 1 a bounded complex consisting of modules in B C (R). Thus we get another triangle: vi [ 1] M I i [ i] in D b (R). Furthermore, we have a triangle: βα Y 1 I i [ i] βα Y 1 M [1] I i [ i + 1] in D b (R). Let Y 2 be the mapping cone of βα. Then there exists an isomorphism M [1] Y 2 in D b (R). Put Y = Y 2 [ 1]. Then Y has finite length and all spots in Y are in B C (R), and so Y is the desired complex. Let Λ be an artin R-algebra over a commutative artin ring R. We denote by D the ordinary Matlis duality, that is, D( ) := Hom R (, E 0 (R/J(R))), where J(R) is the Jacobson radical of R and E 0 (R/J(R)) is the injective envelope of R/J(R). It is easy to verify that (Λ, Λ)-bimodule D(Λ) is semidualizing. Recall that Λ is called Gorenstein if id Λ Λ = id Λ op Λ <. As an application of Theorem 5.5, we get the following Corollary 5.6. Let Λ be an artin algebra. Then the following statements are equivalent for any n 0. (1) Λ is Gorenstein with id Λ Λ = id Λ op Λ n. (2) For any simple module T Mod Λ, B D(Λ) (Λ)-id Λ T n. (3) For any simple module T Mod Λ, there exists a quasi-isomorphism T Y with Y a bounded complex of length at most n + 1 consisting of modules in B D(Λ) (Λ). (4) For any simple module T Mod Λ, there exists an exact sequence: 0 T X T W T 0 in Mod Λ such that X T B D(Λ) (Λ) and id Λ W T n 1. 22

23 (5) For any simple module T Mod Λ, there exists an exact sequence: 0 X T W T T 0 in Mod Λ such that X T B D(Λ) (Λ) and id Λ W T n. Proof. By Theorem 4.2, we have (2) (4) (5). By Theorem 5.5, we have (2) (3). (1) (2) Let T Mod Λ be simple. Since Λ is Gorenstein with id Λ Λ = id Λ op Λ n, it follows from [15, Theorem ] that coω n (T ) is Gorenstein injective. Then coω n (T ) B D(Λ) (Λ) by [33, Corollary 5.2 and Theorem 3.9]. Now the assertion follows from Lemma 5.3. (4) (1) Let T Mod Λ be simple. Then by (4) and [33, Theorem 3.9 and Corollary 4.2], GI(Λ)-id Λ T n. So sup{gp(λ)-pd Λ M M Mod Λ} = sup{gi(λ)- id Λ M M Mod Λ} n by [8, Theorem 1.1] and [32, Theorem 2.1]. It follows from [25, Theorem 1.4] that Λ is Gorenstein with id Λ Λ = id Λ op Λ n. 6. A Dual of Auslander-Bridger s Approximation Theorem In this section, we first obtain a dual version of the Auslander-Bridger s approximation theorem, and then give its several applications. We begin with the following Lemma 6.1. ([36, Proposition 2.2]) (1) For any X Mod R, we have (θ X ) µ X = 1 X. (2) For any Y Mod S, we have θ C S Y (1 C µ Y ) = 1 C S Y. For any n 0, recall from [3] that the grade of a finitely generated R-module M is defined as grade R M := inf{i 0 Ext i R(M, R) 0}; and the strong grade of M, denoted by s.grade R M, is said to be at least n if grade R X n for any submodule X of M. We introduce two dual versions of these notions as follows. Definition 6.2. Let M Mod R, N Mod S and n 0. (1) The Ext-cograde of M with respect to C is defined as E-cograde C M := inf{i 0 Ext i R(C, M) 0}; and the strong Ext-cograde of M with respect to C, denoted by s.e-cograde C M, is said to be at least n if E-cograde C X n for any quotient module X of M. (2) The Tor-cograde of N with respect to C is defined as T-cograde C N := inf{i 0 Tor S i (C, N) 0}; and the strong Tor-cograde of N with respect to C, denoted by s.t-cograde C N, is said to be at least n if T-cograde C Y n for any submodule Y of N. 23

24 We remark that the Tor-cograde of N with respect to C is called the cograde of N with respect to C in [33]. The following result can be regarded as a dual version of the Auslander-Bridger s approximation theorem (see [17, Proposition 3.8]). Theorem 6.3. Let M Mod R and n 1. If T-cograde C Ext i R(C, M) i for any 1 i n, then there exists a module U Mod R and a homomorphism f : U M in Mod R satisfying the following properties: (1) P C (R)-id R U n, and (2) Ext i R(C, f) is bijective for any 1 i n. Proof. We proceed by induction on n. Let n = 1 and f 1 Q 1 Q0 Ext 1 R(C, M) 0 be a projective presentation of Ext 1 R(C, M) in Mod S. Then we get the following exact sequence: 1 C S Q C f 1 1 C S Q 0 C S Ext 1 R(C, M) 0 in Mod R with both C S Q 1 and C S Q 0 in P C (R)(= Add R C). Put U = Ker(1 C f 1 ). Because C S Ext 1 R(C, M) = 0 by assumption, P C (R)-id R U 1. Next we show that there exists a homomorphism f : U M in Mod R such that Ext 1 R(C, f) is bijective. Since Q 1 and Q 0 are projective, there exist two homomorphisms g 0 and g 1 such that we have the following commutative diagram with exact rows: Q 1 f 1 Q0 δ Ext 1 R(C, M) 0 g 1 g 0 I 0 (M) coω 1 (M) δ Ext 1 R(C, M) 0. Diagram (6.1) 24

25 Then there exists a homomorphism f such that we have the following commutative diagram with exact rows: 0 U C S Q 1 1 C f 1 C S Q 0 0 f h 1 0 M I 0 (M) coω 1 (M) 0, h 0 Diagram (6.2) where h 1 = θ I 0 (M) (1 C g 1 ) and h 0 = θ coω 1 (M) (1 C g 0 ). Applying the functor ( ) to Diagram (6.2), we obtain the following commutative diagram with exact rows: (1 C f 1 ) (C S Q 1 ) (C S Q 0 ) δ Ext 1 R(C, U) 0 h 1 h 0 I 0 (M) coω 1 (M) δ Ext 1 R (C,f) Ext 1 R(C, M) 0. Because the following diagram: Diagram (6.3) Q 0 g 0 coω 1 (M) µ Q0 µ coω 1 (M) (1 C g 0 ) (C S Q 0 ) (C S coω 1 (M) ) is commutative, µ coω 1 (M) g 0 = (1 C g 0 ) µ Q0. Then we have h 0 µ Q0 = (θ coω 1 (M) (1 C g 0 )) µ Q0 = (θ coω 1 (M)) (1 C g 0 ) µ Q0 = (θ coω 1 (M)) µ coω 1 (M) g 0 = 1 coω 1 (M) g 0 (by Lemma 6.1(1)) = g 0. On the other hand, from Diagrams (6.1) and (6.3), we get that δ Ext 1 R(C, f) δ = δ h 0. So we have = δ g 0 and Ext 1 R(C, f) δ µ Q0 25

26 = δ h 0 µ Q0 = δ g 0 = δ, and we get the following commutative diagram with exact rows: (C S Q 1 ) (1 C f 1 ) (C S Q 0 ) δ Ext 1 R(C, U) 0 = (µ Q1 ) 1 = (µ Q0 ) 1 Ext 1 R (C,f) Q 1 f 1 Q0 δ Ext 1 R(C, M) 0. Thus Ext 1 R(C, f) is bijective. Now suppose n 2. By the induction hypothesis, there exists a homomorphism f : U M in Mod R such that P C (R)-id R U n 1 and Ext i R(C, f ) is bijective for any 1 i n 1. Then there exists a Hom R (, P C (R))-exact exact sequence: 0 U g W X 0 in Mod R with W in P C (R), and we get the following commutative diagram with exact columns and rows: 0 M ( 1 M 0 ) ( f 0 U ) g M W 0 U g W (0,1 W ) 0 M L X 0 0 where L = Coker ( f 0 0, g ). It is easy to see that the exact sequence: ( f 0 U ) g M W L 0 is Hom R (, P C (R))-exact. Because P C (R)-id R U n 1 and Ext i R(C, f ) is bijective for any 1 i n 1, we have that the sequence 0 U ( f g ) (M W ) L 0 26

27 is exact, Ext 1in 1 R (C, L) = 0 and Ext n R(C, M) = Ext n R(C, L). Take a projective resolution: Q n f n f 2 f 1 Q1 Q0 Ext n R(C, M) 0 (6.1) of Ext n R(C, M) in Mod S. By assumption T-cograde C Ext n R(C, M) n, so we get the following exact sequence: 1 0 N C S Q C f n 1 n C f 2 1 C S Q C f 1 1 C S Q 0 0 (6.2) in Mod R with all C S Q i in P C (R) and N = Ker(1 C f n ). Then P C (R)-id R N n. Applying the functor ( ) to the exact sequence (6.2) we get the following sequence: (1 C f n ) 0 N (C S Q n ) (1 C f 2 ) (1 C f 1 ) (C S Q 1 ) (C S Q 0 ) 0. (6.3) Comparing the sequences (6.1) with (6.3) we get that Ext 1in 1 R (C, N) = 0 and Ext n R(C, N) = Ext n R(C, M). Because Ext i R(C, L) = 0 for any 1 i n 1, we get an exact sequence: I 0 (L) I 1 (L) I n 1 (L) K Ext n R(C, L) 0 in Mod S, where K = Coker(I n 2 (L) I n 1 (L)). Since all Q i are projective, there exist homomorphisms g 0, g 1, g n such that we have the following commutative diagram with exact rows: Q n f n f 2 Q1 f 0 Q0 Ext n R(C, L) 0 g n g 1 I 0 (L) I n 1 (L) K Ext n R(C, L) 0. g 0 Diagram (6.4) Then there exists a homomorphism h such that we have the following commutative diagram with exact rows: 0 N s C S Q n 1 C f n 1 C f 1 C S Q 1 1 C f 0 C S Q 0 0 h h n 0 L I 0 (L) I n 1 (L) K 0, h 1 h 0 Diagram (6.5) 27

28 where h i = θ I n i (L) (1 C g i ) for any 1 i n and h 0 = θ K (1 C g 0 ). Notice that the functor ( ) gives Diagram (6.5) back to Diagram (6.4), so Ext n R(C, h) is bijective. Put W = C S Q n. Then we get an exact sequence: 0 N (h s) L W N 0 and a Hom R (, P C (R))-exact exact sequence: in Mod R, where u = 0 U u M W W L W 0 ) ( f g 0. Consider the following pull-back diagram: U α U β N 0 0 U u M W W λ L W 0 N N 0 0. It is easy to see that the first row in the above diagram is Hom R (, P C (R))-exact exact. Because P C (R)-id R U n 1 and P C (R)-id R N n, P C (R)-id R U n by the dual version of [15, Lemma 8.2.1]. Put p = (1 M, 0, 0) : M W W M and f = p λ. Then Ext i R(C, f) = Ext i R(C, p) Ext i R(C, λ) for any i 0. Because W W P C (R), Ext i R(C, p) is bijective for any i 1. Note that Ext i R(C, f ) is bijective for any 1 i n 1 and Ext 1in 1 R (C, N) = 0 = Ext 1in 1 R (C, L). We have the following commutative diagram with exact rows: Ext i R(C, U ) Ext i R (C,α) Ext i R(C, U) 0 Ext i R (C,λ) 0 Ext i R(C, U Exti R ) (C,u) Ext i R(C, M W W ) 0. So Ext i R(C, λ) and Ext i R(C, f) are bijective for 1 i n 1. On the other hand, because Ext n R(C, h) is bijective and Ext n+1 R (C, U ) = 0 = Ext n 1 (C, L), we have the 28 R

29 following commutative diagram with exact rows: Ext n R(C, U ) Ext n R (C,α) Ext n R(C, U) Ext n R (C,β) Ext n R(C, N) 0 Ext n R (C,λ) 0 Ext n R(C, U Extn R ) (C,u) Ext n R(C, M W W ) Ext n R(C, L W ). = So Ext n R(C, λ) and Ext n R(C, f) are bijective. The proof is finished. Dual to Theorem 6.3, we have the following Theorem 6.4. Let N Mod S and n 1. If E-cograde C Tor S i (C, N) i for any 1 i n, then there exists a module V Mod S and a homomorphism g : N V in Mod S satisfying the following properties: (1) I C (S)-pd S V n, and (2) Tor S i (C, g) is bijective for any 1 i n. In the rest of this section, we give several applications of Theorems 6.3 and 6.4. Let Λ be an artin R-algebra over a commutative artin ring R and mod Λ the class of finitely generated left Λ-modules. It is well known that the ordinary Matlis duality functor D( ) induces a duality between mod Λ and mod Λ op. Recall from [23] that Λ is called a right quasi n-gorenstein algebra provided fd Λ op I i (Λ Λ ) i + 1 for any 0 i n 1. As an application of Theorem 6.3, we get the following Corollary 6.5. Let Λ be a right quasi n-gorenstein artin algebra and M mod Λ. Then there exists a module U mod Λ and a homomorphism f : U M in mod Λ satisfying the following properties: (1) id Λ U n, and (2) Ext i Λ(D(Λ), f) is bijective for any 1 i n. Proof. Let M mod Λ and i, j 0. Then we have Ext i Λ(D(Λ), M) = Ext i Λ(D(Λ), D(D(M))) = D(Tor Λ i (D(M), D(Λ))) (by [9, Chapter VI, Proposition 5.1]) = D(D(Ext i Λop(D(M), Λ))) (by [9, Chapter VI, Proposition 5.3]) = Ext i Λop(D(M), Λ). So for any i 1 and j 0, we have Tor Λ j (D(Λ), Ext i Λ(D(Λ), M)) = Tor Λ j (D(Λ), Ext i Λ op(d(m), Λ)) 29

30 = D(Ext j Λ (Exti Λop(D(M), Λ), Λ) (by [9, Chapter VI, Proposition 5.3]). Since Λ is a right quasi n-gorenstein algebra, grade Λ Ext i Λop(D(M), Λ) i for any 1 i n by [3, Theorem 4.7]. It follows from the above argument that T-cograde D(Λ) Ext i Λ(D(Λ), M) i for any 1 i n. In addition, notice that D(Λ) is an injective cogenerator for Mod Λ, so P D(Λ) (Λ)-id Λ X = id Λ X for any X mod Λ. Now the assertion follows from Theorem 6.3. We give the second application of Theorems 6.3 (and 6.4) as follows. Corollary 6.6. Let M Mod R and N Mod S. Then for any n 0 we have (1) If T-cograde C Ext i R(C, M) i + 1 for any 0 i n, then E-cograde C M n + 1. (2) If E-cograde C Tor S i (C, N) i + 1 for any 0 i n, then T-cograde C N n + 1. Proof. (1) We proceed by induction on n. Let n = 0 and C S M = 0. Since (θ M ) µ M = 1 M by Lemma 6.1(1), µ M is a split monomorphism and M = 0. Now suppose n 1. By the induction hypothesis, we have that E-cograde C M n and Ext 0in 1 R (C, M) = 0. It is left to show Ext n R(C, M) = 0. By Theorem 6.3, there exists a module U Mod R and a homomorphism f : U M in Mod R such that P C (R)-id R U n and Ext i R(C, f) is bijective for any 1 i n. It follows that Ext 1in 1 R (C, U) = 0. Let 0 U g W 0 W 1 W n 0 be an exact sequence in Mod R with all W i in P C (R). Applying the functor ( ) to it we get an exact sequence: 0 U W 0 W 1 W n Ext n R(C, U) 0 in Mod S. Since Ext n R(C, M) = Ext n R(C, U), we have T-cograde C Ext n R(C, U) n+1 by assumption. Then we get the following commutative diagram with exact rows: C R U C S W 0 C S W 1 C S W n 0 θ U θ W0 0 U W 0 W 1 W n θ W1 θ Wn