Lobachevskii Journal of Mathematics

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ISSN 1818-9962

Vol. 26, 2007, 69–77 \begin{matrix}c \\ ◯ \end{matrix} Wei Jiaqun

Wei Jiaqun

A NOTE ON GENERALIZED GORENSTEIN DIMENSION

(submitted by M. M. Arslanov)

Abstract. We prove that two categories G_{ω} and X_{ω}, introduced for the faithfully balanced selforthogonal module ω by Auslander and Reiten in [2] and [3] respectively, coincide with each other. As an application we give a generalization of a main theorem in [6].

1. Introduction

Throughout this note, we assume that R (S respectively) is a left (right respectively) noetherian ring and modules are finitely generated left modules. By a subcategory we mean a full subcategory closed under isomorphisms. We fix ω a faithfully balanced selforthogonal R-module with S = End_{R}ω (i.e. a generalized tilting module in sense of [8]), which is defined to satisfy the conditions (a) R ≃ Endω_{S} and, (b) Ext_{R}^{i}(ω, ω) = 0 = Ext_{S}^{i}(ω, ω) for all i ≥ 1.

The notion of the Gorenstein dimension was introduced in [1] for a twosided noetherian ring R. An R-module M has Gorenstein dimension zero, written G-dimM = 0, if M is a reflexive R-module and Ext_{R}^{i}(M, R) = 0 = Ext_{R}^{i}(M^{∗}, R) for all i ≥ 1, where M^{∗} = Hom_{R}(M, R). Further, G-dimM = n if n is the minimal integer such that there is an exact sequence 0 → L_{n} → ⋯ → L_{0} → M → 0 with G-dimL_{i} = 0 for all 0 ≤ i ≤ n. Otherwise, G-dimM = ∞.

Supported by the NSFC (No.10601024).

In [2], the authors studied a generalization of this dimension. An R-module M is said to be of generalized Gorenstein dimension zero, written G-dim_{ω}M = 0, if Ext_{R}^{i}(M, ω) = 0 = Ext_{S}^{i}(Hom_{R}(M, ω), ω) for all i ≥ 1 and the canonical map σ_{M} : M → Hom_{S}(Hom_{R}(M, ω), ω) is an isomorphism. G-dim_{ω}M = n if n is the minimal integer such that there is an exact sequence 0 → L_{n} → ⋯ → L_{0} → M → 0 with G-dim_{ω}L_{i} = 0 for all 0 ≤ i ≤ n, otherwise, G-dim_{ω}M = ∞. We denoted by G_{ω} the subcategory of R-modules of generalized Gorenstein dimension zero.

It was also introduced in [3] a subcategory, denoted by X_{ω}, whose objects are R-modules M such that there is an exact sequence 0 → M → ω_{0} →^{f_{0}} ω_{1} →^{f_{1}} … with ω_{i} ∈ add_{R}ω, i.e. the subcategory of the direct summands of finite direct sum of ω, with Imf_{j} ∈ ^{⊥}ω = {X ∈ R−mod | Ext_{R}^{i}(X, ω) = 0 for all i ≥ 1} for all j ≥ 0.

One purpose of this note is to study the relation between the abovereferenced two subcategories. It is well known that G_{ω} ⊆ X_{ω} and G_{ω} = X_{ω} when ω is a cotilting bimodule (see [2] for the case of artin algebras and [6] for the case of two-sided noetherian ring). In this note, we will show that G_{ω} = X_{ω} always holds for the faithfully balanced selforthogonal module ω. Note that the cotilting bimodule (in artin algebras [3] and in [6]) is faithfully balanced selforthogonal, but conversely the faithfully balanced selforthogonal module need not be cotilting (see for instance [9, Example 3.1]). As the Gorenstein dimension and its generalizations were recently studied by many mathematicians (e.g. [4, 5, 7] etc.), this result would be helpful for us to understand the (generalized) Gorenstein dimension more precisely.

We in turn compare the generalized Gorenstein dimension with the left orthogonal dimension related to ω defined in [6]. Accordantly, an Rmodule M has left orthogonal dimension related to ω equal to n, written ⊥ω−dimM = n, if n is the minimal integer such that there is an exact sequence 0 → K_{n} → ⋯ → K_{0} → M → 0 with K_{i} ∈ ^{⊥}ω for all 0 ≤ i ≤ n, otherwise, ^{⊥}ω−dimM = ∞. We show that ^{⊥}ω−dimM ≤ G-dim_{ω}M in general and they agree when the latter is finite. Consequently, we get a method to compute the generalized Gorenstein dimension when it is known to be finite. We also study the subcategory of modules of finite generalized Gorenstein dimension, denoted by Ĝ_{ω}, and some known results were extended.

We denote by R−mod (mod−S respectively) the category of all finitely generated left R-modules (right S-modules respectively). Let C be a subcategory of A−mod, we denote by Ĉ the category of all modules M such that there is an exact sequence 0 → C_{m} → ⋯ → C_{0} → M → 0 for some integer m with each C_{i} ∈ C. Assume C ⊃ D are two subcategories of R-mod. Let C ∈ C and D ∈ add_{R}D. A homomorphism D → C is said to be a right D-approximation of C if Hom_{R}(X, D) → Hom_{R}(X, C) is an epimorphism for any X ∈ add_{R}D. D is said to be a contravariantly finite subcategory of C (or D is contravariantly finite in C) if every C ∈ C has a right D-approximation. Dually, A homomorphism C → D is said to be a left D-approximation of C if Hom_{R}(D, X) → Hom_{R}(C, X) is an epimorphism for any X ∈ add_{R}D. D is said to be a covariantly finite subcategory of C (or D is covariantly finite in C) if every C ∈ C has a left D-approximation. D is said to be a functionally finite subcategory of C (or D is functionally finite in C) if D is both a contravariantly finite subcategory and a covariantly finite subcategory of C.

Assume that D is contravariantly finite subcategory in C and D′ is covariantly finite subcategory in C. If every kernel of right D-approximations is in D′, then D′ is called the associated (with D) covariantly finite subcategory. Similarly, if every cokernel of left D′-approximations is in D, then D is called the associated (with D′) contravariantly finite subcategory.

For A ∈ R−mod (mod−S respectively), we put A^{ω} = Hom_{R}(A, ω) (Hom_{S}(A, ω) respectively). An R-module M is said to be ω-reflexive if the natural homomorphism σ_{M} : M → M^{ωω} is an isomorphism.

2. Main Results

Theorem 2.1. G_{ω} = X_{ω}, for the faithfully balanced selforthogonal Rmodule ω.

Proof. It’s sufficient to show X_{ω} ⊆ G_{ω} by [2, Proposition 4.3]. Obviously we have X_{ω} ⊆ ^{⊥}ω. Now let M ∈ X_{ω}. By the definition we see that there exists an exact sequence 0 → M → ω_{0} → N → 0 with ω_{0} ∈ add_{R}ω and N ∈ X_{ω}. Note that N ∈ ^{⊥}ω, so that the induced sequence 0 → N^{ω} → ω_{0}^{ω} → M^{ω} → 0 is exact. Hence we have the following exact commutative diagram, where σ_{N}, σ_{ω_{0}}, σ_{M} are the natural homomorphisms.

\xymatrix{0 \ar[r] & M \ar[r] \ar[d]_{σ_{M}} & ω_{0} \ar[d]_{σ_{ω_{0}}} \ar[r] & N \ar[r] \ar[d]_{σ_{N}} & 0 \\ 0 \ar[r] & M^{ωω} \ar[r] & ω_{0}^{ωω} \ar[r] & N^{ωω} \ar[r] & Ext_{S}^{1}(M^{ω}, ω) \ar[r] & 0}

It is easy to see that σ_{ω_{0}} is a natural isomorphism, so we have that σ_{M} is a monomorphism. Note that N has properties same as M, so σ_{N} is a monomorphism, too. Therefore σ_{M} must be an isomorphism. In the same way we obtain that σ_{N} is an isomorphism. It follows that Ext_{S}^{1}(M^{ω}, ω) = 0. Since Ext_{S}^{i+1}(M^{ω}, ω) ≃ Ext_{S}^{i}(N^{ω}, ω) for i ≥ 1, we see that Ext_{S}^{i}(M^{ω}, ω) = 0 for all i ≥ 1, by repeating the process. Hence M is an R-module of generalized Gorenstein dimension zero and it follows X_{ω} ⊆ G_{ω}. □

Corollary 2.2. G_{ω} is closed under extensions, kernels of epimorphisms and direct summands.

Proof. By Theorem 2.1 and [3, Proposition 5.1] (the proof can be transferred directly to our settings). □

The following proposition compares the generalized Gorenstein dimension with the left orthogonal dimension relative to ω.

Proposition 2.3. Let M ∈ R − mod. Then ^{⊥}ω − dimM ≤ G-dim_{ω}M and they agree when the latter is finite.

Proof. Since G_{ω} ⊆ ^{⊥}ω, we see that ^{⊥}ω − dimM ≤ G-dim_{ω}M by the definitions. Now let G-dim_{ω}M = t < ∞ and ^{⊥}ω − dimM = s. By [6, Lemma 7], s = sup{k | Ext_{R}^{k}(M, ω) ≠ 0}. Hence Ext_{R}^{i}(M, ω) = 0 for all i > s. Consider the sequence 0 → G_{t} →^{f} G_{t−1} → ⋯ → G_{0} → M → 0 with G_{i} ∈ G_{ω} for 0 ≤ i ≤ t. Let N = Cokerf. If t > s strictly, then we have that Ext_{R}^{i}(N, ω) ≃ Ext_{R}^{t−1+i}(M, ω) = 0 for all i ≥ 1. Hence the sequence 0 → N^{ω} → G_{t−1}^{ω} → G_{t}^{ω} → 0 is exact. It follows that 0 → G_{t}^{ωω} → G_{t−1}^{ωω} → N^{ωω} → 0 is also exact and Ext_{S}^{i}(N^{ω}, ω) = 0 for i ≥ 1. Now it is easy to see that N is ω-reflexive. Hence N ∈ G_{ω}. Therefore we have G-dim_{ω}M ≤ t − 1, which is a contradiction. □

Consequently, we get a method to compute the generalized Gorenstein dimension when it is known to be finite.

Corollary 2.4. Assume that G − dim_{ω}M < ∞. Then G − dim_{ω}M = sup{k|Ext_{R}^{k}(M, ω) ≠ 0}.

Proof. By Proposition 2.3 and [6, Lemma 7]. □

The following lemma generalizes [1, Theorem 3.13].

Lemma 2.5. For an integer k ≥ 0 and M ∈ R − mod, the following are equivalent:

(1) G-dim_{ω}Ω_{G_{ω}}^{k}(M) = 0, where Ω_{G_{ω}}^{k}(M) denote k-syzygies of the G_{ω}- resolution of M.

(2) G-dim_{ω}Ω^{k}(M) = 0.

(3) G-dim_{ω}M ≤ k.

Proof. (1)⇒(2). It follows from the fact that G_{ω} contains all finitely generated projective R-modules.

(2)⇒(3). By the definition.

(3)⇒(1). By Corollary 2.2 and the fact that G_{ω} contains all finitely generated projective R-modules, we see that [1, Lemma 3.12] works. It follows that (1) holds. □

Corollary 2.6. Let 0 → A → B → C → 0 be exact in R-mod. If two of A, B, C are in Ĝ_{ω} then the third is also in Ĝ_{ω}. Moreover,

(1) For any n > G-dim_{ω}C, G-dim_{ω}A ≤ n if and only if G-dim_{ω}B ≤ n.

(2) If G-dim_{ω}B = 0 ≠ G-dim_{ω}C, then G-dim_{ω}A = G-dim_{ω}C − 1.

Proof. It’s easy to see that, for any integer t ≥ 1, there is the following exact commutative diagram, where P_{i}, Q_{i}, 0 ≤ i ≤ t − 1, are finitely generated projective R-modules and Ω^{t}(A), Ω^{t}(B), Ω^{t}(C) are some tsyzygies of A, B, C respectively.

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Ω^{t}(A) \ar[d] \ar[r] & Ω^{t}(B) \ar[d] \ar[r] & Ω^{t}(C) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & P_{t−1} \ar[d] \ar[r] & P_{t−1} ⊕ Q_{t−1} \ar[d] \ar[r] & Q_{t−1} \ar[d] \ar[r] & 0 \\  & ⋮ \ar[d] & ⋮ \ar[d] & ⋮ \ar[d] \\ 0 \ar[r] & P_{0} \ar[d] \ar[r] & P_{0} ⊕ Q_{0} \ar[d] \ar[r] & Q_{0} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & A \ar[d] \ar[r] & B \ar[d] \ar[r] & C \ar[d] \ar[r] & 0 \\  & 0 & 0 & 0}

If two of Ω^{t}(A), Ω^{t}(B), Ω^{t}(C) are modules of generalized Gorenstein dimension zero, then the generalized Gorenstein dimension of the third is clearly not more than one. It follows from Lemma 2.5 that, if two of A, B, C in the exact sequence 0 → A → B → C → 0 have finite generalized Gorenstein dimension, then the third also has finite generalized Gorenstein dimension.

For any n > G-dim_{ω}C we easily see that Ω^{n}(C) ∈ G_{ω}. By Corollary 2.2, G_{ω} is closed under extensions and kernels of epimorphisms. Hence Ω^{n}(A) ∈ G_{ω} if and only if Ω^{n}(B) ∈ G_{ω}. It follows again by Lemma 2.5 that G-dim_{ω}A ≤ n if and only if G-dim_{ω}B ≤ n.

Assume that G-dim_{ω}B = 0. If G-dim_{ω}C = 1, then by Lemma 2.5 we have G-dim_{ω}A = 0. If G-dim_{ω}C > 1, then sup{k | Ext_{R}^{k}(C, ω) ≠ 0} = sup{k | Ext_{R}^{k}(A, ω) ≠ 0} + 1 since Ext_{R}^{i+1}(C, ω) ≃ Ext_{R}^{i}(A, ω) for all i ≥ 1. By Proposition 2.3 we see that G-dim_{ω}A = G-dim_{ω}C − 1. □

The following lemma is obvious (cf. [6, Lemma 1]).

Lemma 2.7. For a faithfully balanced selforthogonal R-module ω, it holds that Ext_{R}^{i}(^{⊥}ω, { \atop add_{R}ω}) = 0 for all i ≥ 1.

Lemma 2.8. For any N ∈ Ĝ_{ω}, there exist two exact sequences 0 → L → G → N → 0 and 0 → N → M → G′ → 0 with G, G′ ∈ G_{ω} and L, M ∈ { \atop add_{R}ω} such that G → N is a right G_{ω}-approximation of N and N → M is a left { \atop add_{R}ω}-approximation of N.

Proof. For any N ∈ Ĝ_{ω}, let t(N) = G-dim_{ω}N < ∞. We will proceed by induction on t(N).

If t(N) = 0 i.e., N ∈ G_{ω}, then let L = 0 and G = N, and the first desired exact sequence follows. By Theorem 1, we see that there is an exact sequence 0 → N → ω_{0} → G′ → 0 such that ω_{0} ∈ add_{R}ω and G′ ∈ G_{ω}. It’s easy to see that this is just the second desired exact sequence by Lemma 2.7.

Let t(N) ≥ 1. Consider the exact sequence 0 → M → P_{0} → N → 0 with P_{0} finitely generated projective. By Corollary 2.6, t(M) = t(N) − 1 since P_{0} is clearly in G_{ω}. Therefore, there is an exact sequence 0 → M → B → C → 0 with B ∈ { \atop add_{R}ω} and C ∈ G_{ω}, by the induction assumptions. Now consider the following pushout diagram.

[figure: 0; 0; 0; M; P_{0}; N; 0; 0; B; D; N; 0; C; C; 0; 0]

It follows from the middle column that D ∈ G_{ω}, since G_{ω} is closed under extensions. By Lemma 2.7 we see that the middle row is just the desired first exact sequence. Moreover, note that there is an exact sequence 0 → D → ω′_{0} → A → 0 such that ω′_{0} ∈ add_{R}ω and A ∈ G_{ω}, so we have the following exact commutative diagram.

[figure: 0; 0; 0; B; D; N; 0; 0; B; ω′_{0}; E; 0; A; A; 0; 0]

It follows from the middle row that E ∈ { \atop add_{R}ω}, since B ∈ { \atop add_{R}ω}. By Lemma 2.7 we see that the middle column is just the desired second exact sequence. □

By the above lemma and the definition of homologically finite subcategories, we have the following proposition.

Proposition 2.9. G_{ω} is a contravariantly finite subcategory of Ĝ_{ω} and { \atop add_{R}ω} is the associated covariantly finite subcategory of Ĝ_{ω}.

Lemma 2.10. If for any M ∈ Ĝ_{ω}, M^{ω} (as a right S-module) has finite generalized Gorenstein dimension, then G_{ω} is covariantly finite in Ĝ_{ω}.

Proof. As in Lemma 2.8, we obtain that, for any M^{ω} ∈ mod − S, there exists an exact sequence 0 → A′ → X →^{f} M^{ω} → 0 such that f : X → M^{ω} is a right G_{ω}-approximation of M^{ω} and A′ ∈ { \atop addω}_{S} . Let h = f^{ω} ∘ σ_{M} where σ_{M} : M → M^{ωω} is the canonical homomorphism, we will show that h : M → X^{ω} is a left G_{ω}-approximation of M (note that X^{ω} has generalized Gorenstein dimension zero by the definition).

Let g : M → G be any homomorphism of R-modules with G ∈ G_{ω}. Consider the following diagram.

\xymatrix{M \ar[r]^{σ_{M}} \ar[d]^{g} & M^{ωω} \ar[r]^{f^{ω}} \ar[d]^{g^{ωω}} & X^{ω} \ar[dl]^{ϕ^{ω}} \\ G \ar[r]_{σ_{G}} & G^{ωω}}

By Lemma 2.7, we have Ext_{S}^{1}(G^{ω}, A′) = 0. It follows that Hom_{S}(G^{ω}, X) → Hom_{S}(G^{ω}, M^{ω}) is an epimorphism. Hence there is a homomorphism of right S-modules ϕ : G^{ω} → X such that g^{ω} = f ∘ϕ. Consequently we have g^{ωω} = ϕ^{ω} ∘f^{ω}. Note that σ_{G} is an isomorphism by the definition and that σ_{G} ∘ g = g^{ωω} ∘ σ_{M}, so we have g = σ_{G}^{−1} ∘ g^{ωω} ∘ σ_{M} = σ_{G}^{−1} ∘ ϕ^{ω} ∘ f^{ω} ∘ σ_{M} = σ_{G}^{−1} ∘ ϕ^{ω} ∘ h. It follows that Hom_{R}(X^{ω}, G) → Hom_{R}(M, G) is an epimorphism. Then, by the definition h : M → G^{ω} is a left G_{ω}-approximation of M.

□

Combining Proposition 2.9 and Lemma 2.10, we obtain the following theorem.

Proposition 2.11. Assume the same assumptions as in Lemma 2.10, then G_{ω} is functionally finite in Ĝ_{ω}.

An R-module ω is said to be a cotilting bimodule if it is faithfully balanced selforthogonal and id_{R}ω = idω_{S} < ∞. It is well known that Ĝ_{ω} = R − mod if ω is a cotilting bimodule (see [2] and [6]). Hence we have the following corollary as a special case of theorem 2.11.

Corollary 2.12. [6, Theorem 1] If ω is a cotilting bimodule, then G_{ω} is functionally finite in R − mod.

References

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[2] Auslander M. and Reiten I., Cohen-Macaulay algebras and Gorenstein algebras, Progress in Math. 95 (1991), 221–245.

[3] Auslander M. and Reiten I., Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), 111–152.

[4] Enochs E.E., Jenda O.M.G., Relative Homological Algebra, in: de Gruyter Expositions in Mathematics, Vol. 30, Walter de Gruyter Co., Berlin, 2000.

[5] Holm H. Gorenstein homological dimensions, J. Pure and Appl. Alg., 189(1– 3)(2004), 167–193.

[6] Huang Z., Selforthogonal modules with finite injective dimension, Science in China 43 (2000), 1174–1181.

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[7] Huang Z., On a generalization of the Auslander-Bridger transpose, Comm. Alg. 27(1999), 5791–5812.

[8] Wakamatsu T., on modules with trivial self-extensions, J. Alg. 114 (1988), 106– 114.

[9] Wakamatsu T., Stable equivalence of self-injective algebras and a generalization of tilting modules, J. Alg. 134(1990), 298–325.

Department of Mathematics, Nanjing Normal University, Nanjing 210097, China E-mail address: weijiaqun@njnu.edu.cn

Received March 28, 2007; Revised version May 12, 2007