(Weak) Gorenstein Global Dimension of Semiartinian Rings

Abstract

We prove that if 𝑅 is a semiartinian commutative ring, the Gorenstein global dimension of 𝑅 equals the supremum of the Gorenstein projective and injective dimensions of simple 𝑅-modules, and the weak Gorenstein global dimension of 𝑅 equals the supremum of the Gorenstein flat dimensions of simple 𝑅-modules.

1. Introduction

Throughout the paper, all rings are commutative with identity. Let 𝑅 be a ring, and let 𝑀 be an 𝑅-module. As usual, we use pd_𝑅(𝑀), id_𝑅(𝑀), and fd_𝑅(𝑀) to denote, respectively, the classical projective dimension, injective dimension, and flat dimension of 𝑀.

For a two-sided Noetherian ring 𝑅, Auslander and Bridger [1] introduced the 𝐺-dimension, Gdim_𝑅(𝑀), for every finitely generated 𝑅-module 𝑀. They showed that Gdim_𝑅(𝑀)≤pd_𝑅(𝑀) for all finitely generated 𝑅-modules 𝑀, and equality holds if pd_𝑅(𝑀) is finite.

Several decades later, Enochs and Jenda [2, 3] introduced the notion of Gorenstein projective dimension (𝐺-projective dimension for short), as an extension of 𝐺 dimension to modules that are not necessarily finitely generated, and the Gorenstein injective dimension (𝐺-injective dimension for short) as a dual notion of Gorenstein projective dimension. Then, to complete the analogy with the classical homological dimension, Enochs et al. [4] introduced the Gorenstein flat dimension. Some references are [2–8].

Recall that an 𝑅-module 𝑀 is called Gorenstein projective, if there exists an exact sequence of projective 𝑅-modules 𝐏∶⋯⟶𝑃₁⟶𝑃₀⟶𝑃⁰⟶𝑃¹⟶⋯(1.1)

such that 𝑀≅Im(𝑃₀→𝑃⁰) and such that the functor Hom_𝑅(−,𝑄) leaves 𝐏 exact whenever 𝑄 is a projective 𝑅-module. The complex 𝐏 is called a complete projective resolution.

The Gorenstein injective 𝑅-modules are defined dually.

An 𝑅-module 𝑀 is called Gorenstein flat, if there exists an exact sequence of flat 𝑅-modules: 𝐅∶⋯⟶𝐹₁⟶𝐹₀⟶𝐹⁰⟶𝐹¹⟶⋯(1.2)

such that 𝑀≅Im(𝐹₀→𝐹⁰) and such that the functor 𝐼⊗_𝑅− leaves 𝐅 exact whenever 𝐼 is a right injective 𝑅-module. The complex 𝐅 is called a complete flat resolution.

The Gorenstein projective, injective, and flat dimensions are defined in terms of resolutions and denoted by Gpd(−), Gid(−), and Gfd(−), respectively (see [6, 8, 9]).

In [5], for any associative ring 𝑅, the authors proved the equality supGpd_𝑅(𝑀)∣𝑀isa(left)𝑅-module=supGid_𝑅(𝑀)∣𝑀isa(left)𝑅-module.(1.3) They called the common value of the above quantities the left Gorenstein global dimension of 𝑅 and denoted it by 𝑙.Ggldim(𝑅). Similarly, they set 𝑙.wGgldim(𝑅)=supGfd_𝑅(𝑀)∣𝑀isa(left)𝑅-module,(1.4) which they called the left Gorenstein weak dimension of 𝑅. Since all rings in this paper are commutative, we drop the letter 𝑙.

Recall that an 𝑅-module 𝑀 is called semiartinian, if every nonzero quotient module of 𝑀 has nonzero socle. A ring 𝑅 is said to be semiartinian if it is semiartinian as an 𝑅-module; see [10].

In [11], the authors characterized the (resp., weak) Gorenstein global dimension for an arbitrary associative ring. The purpose of this paper is to apply these characterizations to a commutative semiartinian rings. Hence, we prove that if 𝑅 is a semiartinian commutative ring, the Gorenstein global dimension of 𝑅 equals the supremum of the Gorenstein projective and injective dimension of simple 𝑅-modules (Theorem 2.1), and the weak Gorenstein global dimension of 𝑅 equals the supremum of the Gorenstein flat dimensions of simple 𝑅-modules (Theorem 2.7).

2. Main Results

The first main result of this paper computes the Gorenstein global dimension of semiartinian rings via the Gorenstein projective and injective dimensions of simple modules.

Theorem 2.1. Let 𝑅 be a semiartinian ring and 𝑛 a positive integer. The following conditions are equivalent: (1)Ggldim(𝑅)≤𝑛, (2)sup_𝐶{Gpd_𝑅(𝐶),Gid_𝑅(𝐶)}≤𝑛 where 𝐶 ranges ranges over all simple 𝑅-modules, (3)Ext^𝑅𝑛+1(𝐶,𝑃)=Tor^𝑅𝑛+1(𝐼,𝐶)=0 for all simple 𝑅-modules 𝐶, all projective 𝑅-modules 𝑃, and all injective 𝑅-modules 𝐼. Consequently, Ggldim(𝑅)=sup_𝐶Gpd_𝑅(𝐶),Gid_𝑅(𝐶),(2.1) where 𝐶 ranges ranges over all simple 𝑅-modules.

To prove this theorem, we need the following lemma.

Lemma 2.2 (Theorem 2.1, [11]). Let 𝑅 be a ring and 𝑛 a positive integer. Then, Ggldim(𝑅)≤𝑛 if, and only if, 𝑅 satisfies the following two conditions: (C1):id(𝑃)≤𝑛 for every projective 𝑅-module 𝑃, (C2):pd(𝐼)≤𝑛 for every injective 𝑅-module 𝐼.

Proof of Theorem 2.1. (1)⇒(2) Clear by the definition of Ggldim(𝑅).
(2)⇒(3) By [8, Theorem  2.20], Ext^𝑅𝑛+1(𝐶,𝑃)=0 for all 𝑖>𝑛 and all simple 𝑅-module 𝐶 and all projective module 𝑃 since Gpd_𝑅(𝐶)≤𝑛. Let 𝐼 be an injective 𝑅-module. By [8, Theorem  2.22], Ext^𝑅𝑛+1(𝐼,𝐶)=0 for all 𝑖>𝑛 and all simple 𝑅-module 𝐶 since Gid_𝑅(𝐶)≤𝑛. Then, by [12, Lemma  3.1(1)], Tor^𝑅𝑛+1(𝐼,𝐶)=0 for every simple 𝑅-module 𝐶.
(3)⇒(1) Let 𝑃 be a projective 𝑅-module. By [12, Lemma  4.2(2)], id_𝑅(𝑃)≤𝑛 since Ext^𝑅𝑛+1(𝐶,𝑃)=0 for all simple 𝑅-modules 𝐶. Hence, the condition (C1) of Lemma 2.2 is clear. Let now 𝐼 be an arbitrary injective 𝑅-module. By [12, Lemma  4.2(1)], fd_𝑅(𝐼)≤𝑛 since Tor^𝑅𝑛+1(𝐼,𝐶)=0 for all simple 𝑅-module 𝐶. On the other hand, from [13, Theorem  7.2.5(2) and Corollary  7.2.6(1)⇔(2)], we have (𝑎)∶≤=(𝑏su)p∶=psdu_𝑅p(𝐹p)d∣_𝑅𝐹isaflat𝑅-module≤(𝑐)∶=supid_𝑅(𝑀)∣𝑀isan𝑅-modulewithpd_𝑅(𝑀)<∞(𝑃)∣𝑃isaprojective𝑅-module.(2.2) Moreover, we have just proved that (𝑐)≤𝑛, and so (𝑎)≤𝑛. Accordingly, since fd_𝑅(𝐼)≤𝑛, we have pd_𝑅(𝐼)<∞. Hence, since (𝑏)≤𝑛, we get pd_𝑅(𝐼)≤𝑛. Consequently, the condition (C2) of Lemma 2.2 is clear. As consequence, Ggldim(𝑅)≤𝑛, as desired.

Remark 2.3. From the proof of Theorem 2.1, we can easily see that supid_𝑅(𝑃)∣𝑃isaprojective𝑅-module≤supGpd_𝑅(𝐶)∣𝐶isasimple𝑅-module,(2.3) provided 𝑅 is a semiartinian ring.

Corollary 2.4. Let 𝑅 be a semiartinian ring with finite Gorenstein global dimension. Then, Ggldim(𝑅)=sup_𝐶Gpd_𝑅(𝐶),(2.4) where 𝐶 ranges over all simple 𝑅-modules.

Proof. It is sufficient to prove the inequality Ggldim(𝑅)≤sup_𝐶{Gpd_𝑅(𝐶)}, where 𝐶 ranges over all simple 𝑅-modules. We may suppose 𝑛=sup_𝐶{Gpd_𝑅(𝐶)} finite. By Remark 2.3, sup{id_𝑅(𝑃)∣𝑃isaprojective𝑅-module}≤𝑛. Using [8, Theorem  2.20], Gpd_𝑅(𝑀)≤𝑛 for any 𝑅-module 𝑀. Consequently, Ggldim(𝑅)≤𝑛, as desired.

Recall that ring 𝑅 is called quasi-Frobenius if it is self injective and artinian.

Corollary 2.5. The following conditions are equivalent: (1)𝑅 is quasi-Frobenius, (2)𝑅 is a semiartinian ring and every simple 𝑅-module is Gorenstein projective.

Proof. (1)⇒(2) Follows immediately from [5, Proposition  2.10] and Corollary 2.4.
(2)⇒(1) From Remark 2.3, every projective module is injective. Thus, by [14, Theorem  7.55], 𝑅 is quasi-Frobenius.

Since every perfect ring is semiartinian by [10, Proposition  5.1],we have the following corollary.

Corollary 2.6. If 𝑅 is a perfect ring with Jacobson radical 𝐽, then Ggldim(𝑅)=supGpd_𝑅𝑅𝐽,Gid^𝑅𝑅𝐽.(2.5) Moreover, if 𝑅 is not quasi-Frobenius ring then, Ggldim(𝑅)=1+sup{Gpd_𝑅(𝐽),Gid_𝑅(𝐽)}.

Proof. Using [8, Proposition  2.19] and its dual version, this result is immediate since every simple 𝑅-module is a direct summand of the 𝑅-module 𝑅/𝐽 by [15, Theorem  9.3.4].

The second main result of this paper computes the weak Gorenstein global dimension of semiartinian rings via the Gorenstein flat dimensions of simple modules.

Theorem 2.7. Let 𝑅 be a semiartinian ring and let 𝑛 be a positive integer. The following conditions are equivalent: (1)wGgldim(𝑅)≤𝑛, (2)Gfd_𝑅(𝐶)≤𝑛 for all simple 𝑅-modules 𝐶, (3)Tor^𝑅𝑛+1(𝐼,𝐶)=0 for all simple 𝑅-modules 𝐶 and all injective 𝑅-modules 𝐼.
Consequently, wGgldim(𝑅)=sup_𝐶{Gfd_𝑅(𝐶)}, where 𝐶 ranges ranges over all simple 𝑅-modules.

We need the following lemmas.

Lemma 2.8. Let 𝑀 be an 𝑅-module. If Gfd_𝑅(𝑀)≤𝑛 then Tor^𝑖𝑅(𝐼,𝑀)=0 for all 𝑖>𝑛 and every injective 𝑅-modules 𝐼.

Proof. Using the definition of Gorenstein flat module, the case 𝑛=0 is clear. For 𝑛>0, we consider an 𝑛-step flat resolution of 𝑀, and we use the start case.

Lemma 2.9 (Theorem  2.4, [11]). Let 𝑅 be a ring and 𝑛 a positive integer. The following conditions are equivalent: (1)wGgldim(𝑅)≤𝑛, (2)fd_𝑅(𝐼)≤𝑛 for every injective module 𝐼.

Proof of Theorem 2.7. (1)⇒(2) Follows immediately from the definition of weak Gorenstein global dimension.
(2)⇒(3) Follows from Lemma 2.8.
(3)⇒(1) By [12, Lemma  4.2(1)], fd_𝑅(𝐼)≤𝑛 for every injective 𝑅-module 𝐼 since Tor^𝑅𝑛+1(𝐼,𝐶)=0 for all simple 𝑅-modules 𝐶. Hence, this implication follows from Lemma 2.9.

Recall that a ring is called 𝐼𝐹-ring if every injective module is flat; see [16].

Corollary 2.10. Let 𝑅 be a semiartinian ring. The following are equivalent: (1)𝑅 is an 𝐼𝐹-ring. (2)Every simple 𝑅-module is Gorenstein flat. (3)𝑅 is coherent and 𝐸(𝐶) is flat for every simple 𝑅-module 𝐶 where 𝐸(𝐶) is the injective envelope of 𝐶.

Proof. Using Lemma 2.8, a ring 𝑅 is 𝐼𝐹-ring if and only if wGgldim(𝑅)=0. Hence, the equivalence (1)⇔(2) is an immediate consequence of Theorem 2.7.
(3)⇒(1) Follows from [16, Theorem  3.8].
(1)⇒(3) From Lemma 2.9, if 𝑅 is an 𝐼𝐹-ring, then every 𝑅-module is Gorenstein flat. Then, by [17, Theorem  6], 𝑅 is coherent and self 𝐹𝑃-injective (i.e., Ext^1𝑅(𝑀,𝑅)=0 for every finitely presented 𝑅-module 𝑀). Thus, by [16, Theorem  3.8], 𝐸(𝐶) is flat for every simple 𝑅-module 𝐶, where 𝐸(𝐶) is the injective envelope of 𝐶, and certainly 𝑅 is coherent.

Remark 2.11. Note that the equivalence of (1) and (3) in the above corollary does not need that 𝑅 be semiartinian, see [16, Theorem  3.8] and [18, Proposition  4.2].

Lemma 2.12. Let 0→𝑁→𝑁^→𝑁^→0 be an exact sequence of modules over a coherent ring 𝑅. Then, Gpd_𝑅(𝑁^)≤max{Gpd_𝑅(𝑁^),Gpd_𝑅(𝑁)+1} with equality if Gpd_𝑅(𝑁^)≠Gpd_𝑅(𝑁).

Proof. Using [8, Theorem  3.15] and [8, Theorem  3.14], the proof is similar to that of [19, Corollary  2, page 135].

Corollary 2.13. If 𝑅 is a perfect coherent ring with Jacobson radical 𝐽, then wGgldim(𝑅)=Gfd_𝑅(𝑅/𝐽). Moreover, if 𝑅 is not 𝐼𝐹-ring then, wGgldim(𝑅)=1+sup{Gfd_𝑅(𝐽)}.

Proof. Using [8, Proposition  3.13], the first statement of this result is an immediate consequence of Theorem 2.7 since every simple 𝑅-module is a direct summand of the 𝑅-module 𝑅/𝐽 by [15, Theorem  9.3.4].
Suppose that 𝑅 is not 𝐼𝐹-ring. Then, by Lemma 2.9, wGgldim(𝑅)=Gfd_𝑅(𝑅/𝑁)>0. Therefore, by Lemma 2.12, we deduce from the exact sequence 0⟶𝑁⟶𝑅⟶𝑅𝑁⟶0(2.6) that wGgldim(𝑅)=Gpd_𝑅(𝑅/𝑁)=1+Gpd_𝑅(𝑁).

Acknowledgment

The authors would like to express their sincere thanks to the referees for their helpful suggestions and comments.