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Journal of Algebra Gorenstein syzygy modules
Gorenstein syzygy modules
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Band:
324
Jahr:
2010
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english
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12
DOI:
10.1016/j.jalgebra.2010.10.010
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Journal of Algebra 324 (2010) 3408–3419 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Gorenstein syzygy modules Chonghui Huang a,b , Zhaoyong Huang b,∗ a b Research Institute of Mathematics, University of South China, Hengyang 421001, PR China Department of Mathematics, Nanjing University, Nanjing 210093, PR China a r t i c l e i n f o Article history: Received 26 March 2009 Available online 23 October 2010 Communicated by Eﬁm Zelmanov MSC: 16E05 16E10 Keywords: Gorenstein projective modules Gorenstein nsyzygy modules nsyzygy modules Gorenstein projective dimension Gorenstein transpose a b s t r a c t For any ring R and any positive integer n, we prove that a left Rmodule is a Gorenstein nsyzygy if and only if it is an nsyzygy. Over a left and right Noetherian ring, we introduce the notion of the Gorenstein transpose of ﬁnitely generated modules. We prove that a module M ∈ mod R op is a Gorenstein transpose of a module A ∈ mod R if and only if M can be embedded into a transpose of A with the cokernel Gorenstein projective. Some applications of this result are given. © 2010 Elsevier Inc. All rights reserved. 1. Introduction Throughout this paper, R is an associative ring with identity and Mod R is the category of left Rmodules. In classical homological algebra, the notion of ﬁnitely generated projective modules is an important and fundamental research object. As a generalization of this notion, Auslander and Bridger introduced in [AB] the notion of ﬁnitely generated modules of Gorenstein dimension zero over a left and right Noetherian ring. Over a general ring, Enochs and Jenda introduced in [EJ1] the notion of Gorenstein projective modules (not necessarily ﬁnitely generated). It is well known that these two notions coincide for ﬁnitely generated modules over a left and right Noetherian ring. In particular, Gorenstein projective modules share many nice properties of projective modules (e.g. [AB,C,CFH,CI,EJ1,EJ2,H]). * Corresponding author. Em; ail address: huangzy@nju.edu.cn (Z. Huang). 00218693/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2010.10.010 C. Huang, Z. Huang / Journal of Algebra 324 (2010) 3408–3419 3409 The notion of a syzygy module was deﬁned via the projective resolution of modules as follows. For a positive integer n, a module A ∈ Mod R is called an nsyzygy module (of M) if there exists an exact sequence 0 → A → P n−1 → · · · → P 1 → P 0 → M → 0 in Mod R with all P i projective. Analogously, we call A a Gorenstein nsyzygy module (of M) if there exists an exact sequence 0 → A → G n−1 → · · · → G 1 → G 0 → M → 0 in Mod R with all G i Gorenstein projective. It is trivial that an nsyzygy module is Gorenstein nsyzygy. In Section 2, our main result is that for every n 1, a Gorenstein nsyzygy module is nsyzygy. The following auxiliary proposition plays a crucial role in proving this f main result. Let 0 → A → G 1 −→ G 0 → M → 0 be an exact sequence in Mod R with G 0 and G 1 Gorenstein projective. Then we have the following exact sequences 0 → A → P → G → M → 0 and 0 → A → H → Q → M → 0 in Mod R with P , Q projective and G, H Gorenstein projective. In Section 3, for a left and right Noetherian ring R and a ﬁnitely generated left Rmodule A, we introduce the notion of the Gorenstein transpose of A, which is a Gorenstein version of that of the transpose of A. We establish a relation between a Gorenstein transpose of a module and a transpose of the same module. We prove that a ﬁnitely generated right Rmodule M is a Gorenstein transpose of a ﬁnitely generated left Rmodule A if and only if M can be embedded into a transpose of A with the cokernel Gorenstein projective. Then we give some applications of this result: (1) The direct sum of a ﬁnitely generated Gorenstein projective right Rmodule and a transpose of a ﬁnitely generated left Rmodule A is a Gorenstein transpose of A. (2) For any Gorenstein transpose and any transpose of a ﬁnitely generated left Rmodule, one of them is ntorsionfree if and only if so is the other. (3) A ﬁnitely generated left Rmodule with Gorenstein projective dimension n is a double Gorenstein transpose of a ﬁnitely generated left Rmodule with projective dimension n. 2. Gorenstein syzygy modules Recall from [EJ1] a module G ∈ Mod R is called Gorenstein projective if there exists an exact sequence in Mod R: · · · → P1 → P0 → P 0 → P 1 → · · · , such that: (1) All P i and P i are projective; (2) After applying the functor Hom R ( , P ) the sequence is still exact for any projective module P ∈ Mod R; and (3) G ∼ = Im( P 0 → P 0 ). Let M be a module in Mod R. The Gorenstein projective dimension of M, denoted by Gpd R ( M ), is deﬁned as inf{n  for any exact sequence 0 → G n → · · · → G 1 → G 0 → M → 0 in Mod R with all G i Gorenstein projective}. We have Gpd R ( M ) 0 and we set Gpd R ( M ) inﬁnity if no such integer exists (see [EJ1] or [H]). Lemma 2.1. Let 0 → M 3 → M 2 → M 1 → 0 be an exact sequence in Mod R with M 3 = 0. If M 1 is Gorenstein projective, then Gpd R ( M 3 ) = Gpd R ( M 2 ). Proof. By [H, Theorems 2.24 and 2.20], it is easy to get the assertion. 2 The following result plays a crucial role in this paper. f Proposition 2.2. Let 0 → A → G 1 − → G 0 → M → 0 be an exact sequence in Mod R with G 0 and G 1 Gorenstein projective. Then we have the following exact sequences: 0 → A → P → G → M → 0, and 0 → A → H → Q → M → 0, in Mod R with P , Q projective and G, H Gorenstein projective. 3410 C. Huang, Z. Huang / Journal of Algebra 324 (2010) 3408–3419 Proof. Because G 1 is Gorenstein projective, there exists an exact sequence 0 → G 1 → P → G 2 → 0 in Mod R with P projective and G 2 Gorenstein projective. Then we have the following pushout diagram: 0 0 0 A G1 Im f 0 0 A P B 0 G2 G2 0 0 Consider the following pushout diagram: 0 0 0 Im f G0 M 0 0 B G M 0 G2 G2 0 0 Because both G 0 and G 2 are Gorenstein projective, G is also Gorenstein projective by Lemma 2.1. Connecting the middle rows in the above two diagrams, then we get the ﬁrst desired exact sequence. Since G 0 is Gorenstein projective, there exists an exact sequence 0 → G 3 → Q → G 0 → 0 in Mod R with Q projective and G 3 Gorenstein projective. Dually, taking pullback, one gets the second desired exact sequence. 2 For a positive integer n, recall that a module A ∈ Mod R is called an nsyzygy module (of M) if there exists an exact sequence 0 → A → P n−1 → · · · → P 1 → P 0 → M → 0 in Mod R with all P i projective. Analogously, we give the following Deﬁnition 2.3. For a positive integer n, a module A ∈ Mod R is called a Gorenstein nsyzygy module (of M) if there exists an exact sequence 0 → A → G n−1 → · · · → G 1 → G 0 → M → 0 in Mod R with all G i Gorenstein projective. C. Huang, Z. Huang / Journal of Algebra 324 (2010) 3408–3419 3411 The following theorem is the main result in this section. Theorem 2.4. Let n be a positive integer and 0 → A → G n−1 → G n−2 → · · · → G 0 → M → 0 an exact sequence in Mod R with all G i Gorenstein projective. Then we have the following: (1) There exist exact sequences 0 → A → P n−1 → P n−2 → · · · → P 0 → N → 0 and 0 → M → N → G → 0 in Mod R with all P i projective and G Gorenstein projective. In particular, a module in Mod R is an nsyzygy if and only if it is a Gorenstein nsyzygy. (2) There exist exact sequences 0 → B → Q n−1 → Q n−2 → · · · → Q 0 → M → 0 and 0 → H → B → A → 0 in Mod R with all Q i projective and H Gorenstein projective. Proof. (1) We proceed by induction on n. When n = 1, it has been proved in the proof of Proposition 2.2. Now suppose that n 2 and we have an exact sequence: 0 → A → G n −1 → G n −2 → · · · → G 0 → M → 0 in Mod R with all G i Gorenstein projective. Put K = Coker(G n−1 → G n−2 ). By Proposition 2.2, we get an exact sequence: 0 → A → P n−1 → G n −2 → K → 0 in Mod R with P n−1 projective and G n −2 Gorenstein projective. Put A = Im( P n−1 → G n −2 ). Then we get an exact sequence: 0 → A → G n −2 → G n−3 → · · · → G 0 → M → 0 in Mod R. So, by the induction hypothesis, we get the assertion. (2) The proof is dual to that of (1), so we omit it. 2 For a module M ∈ Mod R, we use pd R ( M ) to denote the projective dimension of M. Corollary 2.5. (See [CFH, Lemma 2.17].) Let M ∈ Mod R and n be a nonnegative integer. If Gpd R ( M ) = n, then there exists an exact sequence 0 → M → N → G → 0 in Mod R with pd R ( N ) = n and G Gorenstein projective. Proof. Let M ∈ Mod R with Gpd R ( M ) = n. Then one uses Theorem 2.4(1) with A = 0 to get an exact sequence 0 → M → N → G → 0 in Mod R with pd R ( N ) n and G Gorenstein projective. By Lemma 2.1, Gpd R ( N ) = n, and thus pd R ( N ) = n. 2 By [H, Theorem 2.20], we have that Gpd R ( M ) n if and only if there exists an exact sequence 0 → G n → P n−1 → · · · → P 1 → P 0 → M → 0 in Mod R with all P i projective and G n Gorenstein projective. The following theorem generalizes this result. In particular, the following theorem was proved by Christensen and Iyengar in [CI, Theorem 3.1] when R is a commutative Noetherian ring. Theorem 2.6. Let M ∈ Mod R and n be a nonnegative integer. Then the following statements are equivalent. (1) Gpd R ( M ) n. (2) For every nonnegative integer t such that 0 t n, there exists an exact sequence 0 → X n → · · · → X 1 → X 0 → M → 0 in Mod R such that X t is Gorenstein projective and X i is projective for i = t. Proof. (2) ⇒ (1) It is trivial. (1) ⇒ (2) We proceed by induction on n. Suppose Gpd R ( M ) 1. Then there exists an exact sequence 0 → G 1 → G 0 → M → 0 in Mod R with G 0 and G 1 Gorenstein projective. By Proposition 2.2 3412 C. Huang, Z. Huang / Journal of Algebra 324 (2010) 3408–3419 with A = 0, we get the exact sequences 0 → P 1 → G 0 → M → 0 and 0 → G 1 → P 0 → M → 0 in Mod R with P 0 , P 1 projective and G 0 and G 1 Gorenstein projective. Now suppose n 2. Then there exists an exact sequence 0 → G n → · · · → G 1 → G 0 → M → 0 in Mod R with G i Gorenstein projective for any 1 i n. Set A = Coker(G 3 → G 2 ). By applying Proposition 2.2 to the exact sequence 0 → A → G 1 → G 0 → M → 0, we get an exact sequence 0 → G n → · · · → G 2 → G 1 → P 0 → M → 0 in Mod R with G 1 Gorenstein projective and P 0 projective. Set N = Coker(G 2 → G 1 ). Then we have Gpd R ( N ) n − 1. By the induction hypothesis, there exists an exact sequence 0 → X n → · · · → X t → · · · → X 1 → P 0 → M → 0 in Mod R such that P 0 is projective and X t is Gorenstein projective and X i is projective for i = t and 1 t n. Now we need only to prove (2) for t = 0. Set B = Coker(G 2 → G 1 ). By the induction hypothesis, we get an exact sequence 0 → P n → · · · → P 3 → P 2 → G 1 → B → 0 in Mod R with G 1 Gorenstein projective and P i projective for any 2 i n. Set C = Coker( P 3 → P 2 ). Then by applying Proposition 2.2 to the exact sequence 0 → C → G 1 → G 0 → M → 0, we get an exact sequence 0 → C → P 1 → G 0 → M → 0 in Mod R with P 0 projective and G 0 Gorenstein projective. Thus we obtain the desired exact sequence 0 → P n → · · · → P 2 → P 1 → G 0 → M → 0. 2 Let X be a full subcategory of Mod R. Recall from [EJ2] that a morphism f : X → M in Mod R Hom R ( X , f ) −−−−−−−→ Hom R ( X , M ) → 0 is exact with X ∈ X is called an X precover of M if Hom R ( X , X ) − for any X ∈ X . We use G P( R ) to denote the full subcategory of Mod R consisting of Gorenstein projective modules. Let M ∈ Mod R with Gpd R ( M ) = n < ∞. Taking t = 0 in Theorem 2.6, one gets an exact sequence 0 → N → G → M → 0 in Mod R with G Gorenstein projective and pd R ( N ) n − 1. It is easy to see that this exact sequence is a surjective G P( R )precover of M (see [H, Theorem 2.10]). Remark 2.7. It is known that a module A ∈ Mod R is called an ncosyzygy module (of M) if there exists an exact sequence 0 → M → I 0 → I 1 → · · · → I n−1 → A → 0 in Mod R with all I i injective. Recall from [EJ1] that a module E ∈ Mod R is called Gorenstein injective if there exists an exact sequence in Mod R: · · · → I1 → I0 → I0 → I1 → · · · , such that: (1) All I i and I i are injective; (2) After applying the functor Hom R ( I , ) the sequence is still exact for any injective module I ∈ Mod R; and (3) E ∼ = Im( I 0 → I 0 ). We call A a Gorenstein ncosyzygy module (of M) if there exists an exact sequence 0 → M → E 0 → E 1 → · · · → E n−1 → A → 0 in Mod R with all E i Gorenstein injective. We point out the dual versions on Gorenstein injectivity and (Gorenstein) ncosyzygy of all of the above results also hold true by using completely dual arguments. 3. Gorenstein transpose In this section, R is a left and right Noetherian ring and mod R is the category of ﬁnitely generated left Rmodules. For any A ∈ mod R, there exists a projective presentation in mod R: f P1 − → P 0 → A → 0. Then we get an exact sequence ∗ f 0 → A ∗ → P 0∗ −→ P 1∗ → Coker f ∗ → 0 in mod R op , where ( )∗ = Hom( , R ). Recall from [AB] that Coker f ∗ is called a transpose of A, and denoted by Tr A. We remark that the transpose of A depends on the choice of the projective presentation of A, but it is unique up to projective equivalence (see [AB]). C. Huang, Z. Huang / Journal of Algebra 324 (2010) 3408–3419 3413 Analogously, we introduce the notion of Gorenstein transpose of modules as follows. Let A ∈ mod R. Then there exists a Gorenstein projective presentation in mod R: g π : X 1 −→ X 0 → A → 0, and we get an exact sequence: ∗ g 0 → A ∗ → X 0∗ − −→ X 1∗ → Coker g ∗ → 0 in mod R op . We call Coker g ∗ a Gorenstein transpose of A, and denote it by Trπ G A. It is trivial that a transpose of A is a Gorenstein transpose of A, but the converse does not hold true in general. For example, for a module A in mod R, if A is Gorenstein projective but not projective, then some Gorenstein transpose of A is zero, and any transpose of A is Gorenstein projective (see Proposition 3.4(3) below) but nonzero (otherwise, if a transpose of A is zero, then A is projective, which is a contradiction). Let A ∈ mod R. Recall from [AB] that A is said to have Gorenstein dimension zero if ExtiR ( A , R ) = 0 = ExtiR op (Tr A , R ) for any i 1. It is easy to see that if A has Gorenstein dimension zero, then so does A ∗ . In addition, it is well known that A has Gorenstein dimension zero if and only if it is Gorenstein projective. Let σ A : A → A ∗∗ deﬁned via σ A (x)( f ) = f (x) for any x ∈ A and f ∈ A ∗ be the canonical evaluation homomorphism. Recall that a module A ∈ mod R is called torsionless (resp. reﬂexive) if σ A is a monomorphism (resp. an isomorphism) The following result establishes a relation between a Gorenstein transpose of a module with a transpose of the same module. Theorem 3.1. Let M ∈ mod R op and A ∈ mod R. Then M is a Gorenstein transpose of A if and only if M can be embedded into a transpose Tr A of A with the cokernel Gorenstein projective, that is, there exists an exact sequence 0 → M → Tr A → H → 0 in mod R op with H Gorenstein projective. Proof. We ﬁrst prove the necessity. Assume that M (∼ = TrπG A ) is a Gorentein transpose of A. Then g there exists an exact sequence π : X 1 − −→ X 0 → A → 0 in mod R with X 0 and X 1 Gorenstein projec ∗ tive such that Trπ G A = Coker g . So there exists an exact sequence 0 → H 1 → P 0 → X 0 → 0 in mod R with P 0 projective and H 1 Gorenstein projective. Let K 1 = Im g and g = i α be the natural epicmonic decomposition of g. Then we have the following pullback diagram: 0 0 H 1 H 1 0 K 1 P 0 A 0 0 K1 X0 A 0 0 i 0 3414 C. Huang, Z. Huang / Journal of Algebra 324 (2010) 3408–3419 Now consider the following pullback diagram: 0 0 H 1 H 1 K 1 0 K1 0 0 K2 G 0 K2 X1 α 0 0 where K 2 = Ker g. Because both X 1 and H 1 are Gorenstein projective, G is Gorenstein projective by Lemma 2.1. So there exists an exact sequence 0 → G 1 → P 0 → G → 0 in mod R with P 0 projective and G 1 Gorenstein projective. Consider the following pullback diagram: 0 0 0 G1 G1 K 2 P0 K 1 0 K2 G K 1 0 0 0 P0 K 1 0 K 1 0 K1 0 β 0 So we get the following commutative diagram with exact rows: 0 K 2 β 0 K2 G 0 K2 X1 α C. Huang, Z. Huang / Journal of Algebra 324 (2010) 3408–3419 3415 It yields the following commutative diagram with exact columns and rows: 0 0 0 Ker β H1 H 1 K 2 P0 K 1 0 K1 0 0 β 0 K2 X1 0 0 α 0 h where H 1 = Ker( P 0 → X 1 ). By the snake lemma, we get the exact sequence 0 → Ker β → H 1 −→ H 1 → 0. By Lemma 2.1, H 1 is Gorenstein projective and hence Ker β is also Gorenstein projective. Combining the above diagram with the ﬁrst one in this proof, we get the following commutative diagram with exact columns and rows: 0 0 0 0 Ker β H1 0 K 2 P0 h H 1 0 P 0 A 0 X0 A 0 β 0 K2 X1 0 0 g 0 By applying the functor ( )∗ to the above diagram, we get the following commutative diagram with exact columns and rows: 0 H 1∗ 0 h∗ P 0∗ X 1∗ 0 H 1 0 ∗ A∗ 0 X 0∗ A∗ 0 P 0 g∗ ∗ 0 3416 C. Huang, Z. Huang / Journal of Algebra 324 (2010) 3408–3419 By the snake lemma, we get an exact sequence: ∗ → Tr A → Coker h∗ → 0 0 → Trπ G A = Coker g in mod R op with Coker h∗ (∼ = (Ker h)∗ ∼ = (Ker β)∗ ) Gorenstein projective. f We next prove the suﬃciency. Let P 1 −→ P 0 → A → 0 be a projective presentation of A in mod R. Then we have the following pullback diagram: 0 0 A∗ A∗ P 0∗ P 0∗ f∗ h 0 P 1∗ H 0 M Tr A H 0 0 0 K g 0 Because H is Gorenstein projective and P 1∗ is projective, K is Gorenstein projective by Lemma 2.1. Again because H is Gorenstein projective, by applying the functor ( )∗ to the above commutative diagram, we get the following commutative diagram with exact columns and rows: 0 H∗ 0 0 (Tr A )∗ M∗ 0 g∗ 0 H∗ P 1∗∗ K∗ f ∗∗ P 0∗∗ 0 h∗ P 0∗∗ A 0 By the snake lemma, we have Im h∗ ∼ = Im f ∗∗ . Thus we get Coker h∗ = P 0∗∗ / Im h∗ ∼ = P 0∗∗ / Im f ∗∗ ∼ = A, and therefore we get a Gorenstein projective presentation of A in mod R: h∗ K ∗ −→ P 0∗∗ → A → 0. C. Huang, Z. Huang / Journal of Algebra 324 (2010) 3408–3419 3417 h∗∗ Because both K and P 0∗ are reﬂexive, we get an exact sequence 0 → A ∗ → P 0∗∗∗ − −−→ K ∗∗ → M → 0 in mod R op and M is a Gorenstein transpose of A. 2 As a consequence of Theorem 3.1, we get the following Corollary 3.2. Let A ∈ mod R. Then for any Gorenstein projective module H ∈ mod R op and any transpose Tr A of A, H ⊕ Tr A is a Gorenstein transpose of A. Proof. Assume that H ∈ mod R op is a Gorenstein projective module. Then there exists an exact sequence 0 → H → P → H → 0 in mod R op with P projective and H Gorenstein projective, which induces an exact sequence 0 → H ⊕ Tr A → P ⊕ Tr A → H → 0. Because P ⊕ Tr A is again a transpose of A, H ⊕ Tr A is a Gorenstein transpose of A by Theorem 3.1. 2 It is clear that the Gorenstein transpose of a module A in mod R depends on the choice of the Gorenstein projective presentation of A. Corollary 3.2 provides a method to construct a Gorenstein transpose of a module from a transpose of the same module. It is interesting to ask the following Question 3.3. Is any Gorenstein transpose obtained in this way? If the answer to this question is positive, then we can conclude that the Gorenstein transpose of a module is unique up to Gorenstein projective equivalence. Let A ∈ mod R. By [A, Proposition 6.3] (or [AB, Proposition 2.6]), there exists an exact sequence: σA 0 → Ext1R op (Tr A , R ) → A −−→ A ∗∗ → Ext2R op (Tr A , R ) → 0 (∗) in mod R. For a positive integer n, recall from [AB] that A is called ntorsionfree if ExtiR op (Tr A , R ) = 0 for any 1 i n. From the exact sequence (∗), it is easy to see that A is torsionless (resp. reﬂexive) if and only if it is 1torsionfree (resp. 2torsionfree). The following result shows that some homological properties of any Gorenstein transpose and any transpose of a given module are identical. Proposition 3.4. Let A ∈ mod R. Then for any Gorenstein transpose Trπ G A and any transpose Tr A of A, we have i ∼ (1) ExtiR op (Trπ G A , R ) = Ext R op (Tr A , R ) for any i 1. A is ntorsionfree if and only if so is Tr A. (2) For any n 1, Trπ G (3) Some Gorenstein transpose of A is zero if and only if A is Gorenstein projective, if and only if any (Gorenstein) transpose of A is Gorenstein projective. (4) Gpd R op (Trπ G A ) = Gpd R op (Tr A ). Proof. (1) It is an immediate consequence of Theorem 3.1. (2) Let Trπ G A be any Gorenstein transpose of A. By Theorem 3.1, there exists a transpose Tr A of A op with H Gorenstein projective. satisfying the exact sequence 0 → Trπ G A → Tr A → H → 0 in mod R If Ext1R (Tr(Tr A ), R ) = 0, then Tr A is torsionless. So Trπ is also torsionless and G A Ext1R (Tr(Trπ A ), R ) = 0. Because H is Gorenstein projective, we get an exact sequence 0 → G Tr H → Tr(Tr A ) → Tr(Trπ A ) → 0 in mod R with Tr H Gorenstein projective. So we have that G i 1 π A ), R ) → Ext1 (Tr(Tr A ), R ) → 0 ∼ A ), R ) Ext ( Tr ( Tr A ), R ) for any i 2, and Ext ( Tr ( Tr ExtiR (Tr(Trπ = R R R G G i is exact. So for any i 1, ExtiR (Tr(Trπ G A ), R ) = 0 if and only if Ext R (Tr(Tr A ), R ) = 0, and thus we conclude that for any n 1, Trπ G A is ntorsionfree if and only if so is Tr A. (3) Because A is a (Gorenstein) transpose of any (Gorenstein) transpose of A, it is not diﬃcult to verify the assertion by (1) and (2). 3418 C. Huang, Z. Huang / Journal of Algebra 324 (2010) 3408–3419 π (4) Let Trπ G A be any Gorenstein transpose of A. If TrG A = 0, then the assertion follows from (3). Now suppose Trπ A = 0. By Theorem 3.1, there exists a transpose Tr A of A satisfying the exact seG op with H Gorenstein projective. Then we have that quence 0 → Trπ G A → Tr A → H → 0 in mod R Gpd R op (Trπ G A ) = Gpd R op (Tr A ) by Lemma 2.1. 2 Let A ∈ mod R. By Proposition 3.4(1), we have that A is ntorsionfree if and only if π ExtiR op (Trπ G A , R ) = 0 for any (or some) Gorenstein transpose TrG A of A and 1 i n. On the other hand, also by Proposition 3.4(1), we get a Gorenstein version of the formula (∗) as follows. For any Gorenstein transpose Trπ G A of A, we have the following exact sequence: σA 0 → Ext1R op Trπ −→ A ∗∗ → Ext2R op TrπG A , R → 0 G A, R → A − in mod R. It is easy to see that A is a Gorenstein transpose of Trπ G A. So we also get the following exact sequence: σTrπ A π 0 → Ext1R ( A , R ) → Trπ G A −−−→ TrG A G ∗∗ → Ext2R ( A , R ) → 0 in mod R op . The following result shows that any double Gorenstein transpose of A shares some homological properties of A. Corollary 3.5. Let A ∈ mod R. Then for any Gorenstein transpose Trπ G A of A and any Gorenstein transpose π π Trπ G (TrG A ) of TrG A, we have π i ∼ (1) ExtiR (Trπ G (TrG A ), R ) = Ext R ( A , R ) for any i 1. π (2) For any n 1, Trπ G (TrG A ) is ntorsionfree if and only if so is A. π (3) Gpd R (Trπ G (TrG A )) = Gpd R ( A ). Proof. Note that A is a Gorenstein transpose of any Gorenstein transpose Trπ G A of A. So all of the assertions follow from Proposition 3.4. 2 Note that a transpose of a module is a special Gorenstein transpose of the same module. The following result shows that a module with Gorenstein projective dimension n is a double Gorenstein transpose of a module with projective dimension n. Proposition 3.6. Let A ∈ mod R and n be a nonnegative integer. Then Gpd R ( A ) = n if and only if there exists a module B ∈ mod R with pd R ( B ) = n such that A is a Gorenstein transpose of some transpose Tr B of B (that π is, A = Trπ G (Tr B ), where TrG (Tr B ) is a Gorenstein transpose of some transpose Tr B of B). Proof. Assume that A ∈ mod R with Gpd R ( A ) = n. By Corollary 2.5, there exists an exact sequence 0 → A → B → H → 0 in mod R with pd R ( B ) = n and H Gorenstein projective. Note that B is a transpose of some transpose Tr B of B. By Theorem 3.1, A is a Gorenstein transpose of Tr B. Conversely, if A is a Gorenstein transpose of some transpose Tr B of a module B ∈ mod R with pd R ( B ) = n, then Gpd R ( A ) = Gpd R ( B ) = pd R ( B ) = n by Corollary 3.5. 2 Acknowledgments This research was partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education, NSFC (Grant No. 10771095) and NSF of Jiangsu Province of China (Grant Nos. BK2010047, BK2010007). 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