Co-Cohen-Macaulay Modules and Local Cohomology

Abstract

Let (𝑅,𝔪) be a commutative Noetherian local ring and let 𝑀 be a finitely generated 𝑅-module of dimension 𝑑. Then the following statements hold: (a) if width (𝐻^𝑖_𝔪(𝑀))≥𝑖−1 for all 𝑖 with 2≤𝑖<𝑑, then 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑; (b) if 𝑀 is an unmixed 𝑅-module and depth 𝑀≥𝑑−1, then 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑 if and only if 𝐻_𝔪^𝑑−1(𝑀) is either zero or co-Cohen-Macaulay of Noetherian dimension 𝑑−2. As consequence, if 𝐻^𝑖_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑖 for all 𝑖 with 0≤𝑖<𝑑, then 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑.

1. Introduction

Throughout this paper, let (𝑅,𝔪) be a commutative Noetherian local ring and let 𝑀 be a finitely generated 𝑅-module of dimension 𝑑. We denote the 𝑖th local cohomology module of 𝑀 with respect to 𝔪 by 𝐻^𝑖_𝔪(𝑀). It is well known that 𝐻^𝑖_𝔪(𝑀) is Artinian for all 𝑖 (cf. [1]).

The Noetherian dimension of an Artinian 𝑅-module 𝐴, denoted by 𝑁-dim𝐴, is defined inductively as follows: when 𝐴=0, put 𝑁-dim𝐴=−1. Then by induction, for any integer 𝑟≥0, put 𝑁-dim𝐴=𝑟 if 𝑁-dim𝐴<𝑟 is false, and for any ascending chain 𝐴₀⊆𝐴₁⊆𝐴₂⊆⋯ of submodules of 𝐴 there exists an integer 𝑛₀ such that 𝑁-dim(𝐴_𝑛+1/𝐴_𝑛)<𝑟 for all 𝑛>𝑛₀. Therefore 𝑁-dim𝐴=0 if and only if 𝐴 is a nonzero Noetherian module. Moreover, if 0→𝐴^′→𝐴→𝐴^′′→0 is an exact sequence of Artinian modules, then 𝑁-dim𝐴=max{𝑁-dim𝐴^′,𝑁-dim𝐴^′′}. Let 𝑥₁,…,𝑥_𝑛∈𝔪. 𝑥₁,…,𝑥_𝑛 is an 𝐴-coregular sequence if 0∶_𝐴(𝑥₁,…,𝑥_𝑖−1)𝑅−→^𝑥𝑖0∶_𝐴(𝑥₁,…,𝑥_𝑖−1)𝑅 is surjective for 𝑖=1,…,𝑛 and 0∶_𝐴(𝑥₁,…,𝑥_𝑛)𝑅≠0. The width of 𝐴, denoted by width(𝐴), is the length of any maximal 𝐴-coregular sequence in 𝔪. For any 𝐴-coregular element 𝑥∈𝔪, we have that 𝑁-dim(0∶_𝐴𝑥)=𝑁-dim𝐴−1 and width(0∶_𝐴𝑥)=width𝐴−1. Details about 𝑁-dim𝐴 and width𝐴 can be found in Roberts [2], Kirby [3], and Ooishi [4]; there is a general fact: for any Artinian 𝑅-module 𝐴width𝐴≤𝑁-dim𝐴<∞ holds and 𝐴 is co-Cohen-Macaulay if and only if width𝐴=𝑁-dim𝐴 holds (cf. [5–7]). Tang [8] has shown that if either 𝑑≤2 or 𝑀 is Cohen-Macaulay, then 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay (see also [9]). Following Nagata [10], 𝑀 is unmixed if dim(𝑅/𝔭)=𝑑 for all 𝑅𝔭∈Ass𝑀.

The main aim of this paper is to prove the following theorem.

Theorem 1. The following statements are true.(a)If width(𝐻^𝑖_𝔪(𝑀))≥𝑖−1 for all 𝑖 with 2≤𝑖<𝑑, then 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑. (b)If 𝑀 is an unmixed 𝑅-module and depth𝑀≥𝑑−1, then 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑 if and only if 𝐻_𝔪^𝑑−1(𝑀) is either zero or co-Cohen-Macaulay of Noetherian dimension 𝑑−2.

2. The Results

Following Macdonald [11], every Artinian 𝑅-module 𝐴 has minimal secondary representation 𝐴=𝐴₁+⋯+𝐴_𝑛, where 𝐴_𝑖 is 𝔭_𝑖 secondary. The set {𝑝₁,…,𝑝_𝑛} is independent of the choice of the minimal secondary representation of 𝐴. This set is called the set of attached prime ideals of 𝐴 and denoted by Att𝐴. The set of all minimal elements of Att𝐴 is exactly the set of all minimal elements of Var(Ann𝐴). A sequence (𝑥₁,…,𝑥_𝑛) of elements in 𝔪 is called a strict f-sequence of 𝑀 if 𝑥_𝑗+1∉𝔭 for all 𝔭∈∪^𝑑−𝑗_𝑖=1Att(𝐻^𝑖_𝔪(𝑀/(𝑥₁,…,𝑥_𝑗)𝑀)⧵{𝔪} for all 𝑗=0,1,…,𝑛−1. This notion was introduced in [12].

Lemma 2 (see [9]). For all integer 0≤𝑖<𝑑, one has 𝑁-dim(𝐻^𝑖_𝔪(𝑀))≤𝑖 and 𝑁-dim(𝐻^𝑑_𝔪(𝑀))=𝑑.

Lemma 3. Let 𝑥∈𝔪 be a strict 𝑓-sequence of 𝑀. Then the following statements are true.(i)Suppose that 𝑑≥3 and 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑. Then 𝐻_𝔪^𝑑−1(𝑀/𝑥𝑀) is also co-Cohen-Macaulay of Noetherian dimension 𝑑−1. (ii)Suppose that 𝐻_𝔪^𝑑−1(𝑀/𝑥𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑−1 and width(𝐻^𝑑_𝔪(𝑀))≥3. Then 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑.

Proof. (i) By our hypothesis and using [1, Exercise 11.3.9], we have ℓ(0∶_𝑀𝑥)<∞. Hence from the exact sequences 0⟶0∶_𝑀𝑥𝑀⟶𝑀⟶0∶_𝑀𝑥𝑀⟶0,0⟶0∶_𝑀𝑥^𝑥𝑀⟶𝑀⟶𝑥𝑀⟶0, (1) we get the exact sequence 𝐻0⟶^𝑖𝔪(𝑀)𝑥𝐻^𝑖_𝔪(𝑀)⟶𝐻^𝑖_𝔪𝑀⟶𝑥𝑀0∶_𝐻𝔪^𝑖+1_(𝑀)𝑥⟶0(†), (2) for all 𝑖=1,2,…,𝑑−1. Thus in case 𝑖=𝑑−1 we have that (0∶_𝐻^𝑑_𝔪(𝑀)𝑥) is co-Cohen-Macaulay of Noetherian dimension 𝑑−1. By the choice of 𝑥∈𝔪 the module 𝐻_𝔪^𝑑−1(𝑀)/𝑥𝐻_𝔪^𝑑−1(𝑀) is an 𝑅-module of finite length. Moreover, since width(𝐻_𝔪^𝑑−1(𝑀/𝑥𝑀))≥min{2,𝑑−1}>0 by [8, Proposition 2.4], we have 𝐻^𝔪₀(𝐻_𝔪^𝑑−1(𝑀/𝑥𝑀))=0 by [13, Theorem 4.11]. Thus, by [13, Corollary 3.7], the long exact local homology sequence with respect to 𝑖=𝑑−1 over the exact sequence (†) provides 𝐻_𝔪^𝑑−1(𝑀)/𝑥𝐻_𝔪^𝑑−1(𝑀)=0. Hence there is an isomorphism 𝐻_𝔪^𝑑−1(𝑀/𝑥𝑀)≅(0∶_𝐻^𝑑_𝔪(𝑀)𝑥) and so 𝐻_𝔪^𝑑−1(𝑀/𝑥𝑀) is a co-Cohen-Macaulay module of Noetherian dimension 𝑑−1.
The proof of (ii) follows by the same arguments as in the proof of (i).

Brodmann and Sharp [14], for all integer 𝑖≥0, defined the set {𝔭∈Spec(𝑅)∶𝐻^{𝑖−dim𝑅/𝔭}_𝔭𝑅𝔭(𝑀_𝔭)≠0}, the 𝑖th pseudo support of 𝑀, and denoted by Psupp^𝑖(𝑀). Note that if 𝑅 is complete with respect to 𝔪-adic topology, then by [15, Theorem 3.1] Var(Ann(𝐻^𝑖_𝔪(𝑀)))=Psupp^𝑖(𝑀). The module 𝑀 satisfies Serre's condition 𝑆_𝑛, where 𝑛 is nonnegative integer, provided depth𝑀_𝔭≥min{𝑛,dim𝑀_𝔭} for all 𝔭∈Supp(𝑀). Note that 𝑀 satisfies the condition Serre 𝑆₁ if and only if 𝑀 has no imbedded primes, that is, Ass𝑀=minAss𝑀 is unmixed.

Lemma 4. Let 𝑀 be unmixed and 𝑑≤2. Then 𝐻_𝔪^𝑑−1(𝑀) is either zero or co-Cohen-Macaulay of Noetherian dimension 𝑑−2.

Proof. When 𝑑=0, it is trivial. We assume that 𝑑=1. Since 𝑀 is unmixed then depth𝑀≥1 and so we have 𝐻⁰_𝔪(𝑀)=0. Thus the result has been proved in this case. Now assume that 𝑑=2 and 𝐻¹_𝔪(𝑀)≠0. By using [6, Theorem 1.4], we can assume that 𝑅 is complete with respect to 𝔪-adic topology. Let 𝔭∈Psupp¹(𝑀). Then 𝐻^{1−dim𝑅/𝔭}_𝔭𝑅𝔭(𝑀_𝔭)≠0. Therefore 1≤depth𝑀_𝔭≤1−dim𝑅/𝔭 and so dim𝑅/𝔭=0. Hence Psupp¹(𝑀)={𝔪} and so by [15, Theorem 3.1] Att(𝐻¹_𝔪(𝑀))⊆{𝔪}. This implies that 𝐻¹_𝔪(𝑀) is of finite length (see [1, Corollary 7.2.12]). Hence 𝐻¹_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension zero.

Theorem 5. Let 𝑀 be an unmixed 𝑅-module. If depth𝑀≥𝑑−1, then the following statements are equivalent: (i)the module 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑; (ii)the module 𝐻_𝔪^𝑑−1(𝑀) is either zero or co-Cohen-Macaulay of Noetherian dimension 𝑑−2.

Proof. (i) ⇒ (ii). We use induction on 𝑑. The case 𝑑≤2 follows by Lemma 4. Let 𝑑≥3. Let 𝑥∈𝔪 be a strict 𝑓-sequence on 𝑀. By [1, Exercise 11.3.9], 𝑥∉𝔭 for all 𝔭∈Ass𝑀⧵{𝔪}. Note that 𝔪∉Ass𝑀, since depth𝑀>0. Therefore 𝑥 is 𝑀 regular. Thus, by Lemma 3, 𝐻_𝔪^𝑑−1(𝑀/𝑥𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑−1. Since 𝑀/𝑥𝑀 unmixed and depth𝑀/𝑥𝑀≥𝑑−2, it follows from the inductive hypothesis that 𝐻_𝔪^𝑑−2(𝑀/𝑥𝑀) is either zero or co-Cohen-Macaulay of Noetherian dimension 𝑑−3. Hence, from the exact sequence 𝐻0⟶^𝔪𝑑−2(𝑀)𝑥𝐻_𝔪^𝑑−2(𝑀)⟶𝐻_𝔪^𝑑−2𝑀⟶𝑥𝑀0∶_𝐻𝔪^𝑑−1_(𝑀)𝑥⟶0, (3) and our assumption we get 𝐻_𝔪^𝑑−2(𝑀/𝑥𝑀)≅(0∶_𝐻𝔪^𝑑−1_(𝑀)𝑥) and so (0∶_𝐻𝔪^𝑑−1_(𝑀)𝑥) is either zero or co-Cohen-Macaulay of Noetherian dimension 𝑑−3. Since 𝑥 is a coregular sequence on 𝐻_𝔪^𝑑−1(𝑀), we have 𝐻_𝔪^𝑑−1(𝑀) being either zero or co-Cohen-Macaulay of Noetherian dimension 𝑑−2, as required.
(ii) ⇒ (i). We prove by induction on 𝑑. By [8, Corollary 2.5], we can assume that 𝑑≥3. By our hypothesis there exists 𝑥∈𝔪⧵(∪_{𝔭∈Ass(𝑀)}𝔭)∪(∪_{𝔮∈Att(𝐻𝔪}^𝑑−1_(𝑀))𝔮). Hence, from the exact sequence 𝐻0⟶^𝑖𝔪(𝑀)𝑥𝐻^𝑖_𝔪(𝑀)⟶𝐻^𝑖_𝔪𝑀⟶𝑥𝑀0∶_𝐻𝔪^𝑖+1_(𝑀)𝑥⟶0, (4) we have the isomorphism 𝐻^𝑖_𝔪(𝑀/𝑥𝑀)≅(0∶_𝐻𝔪^𝑖+1_(𝑀)𝑥)(‡) for 𝑖∈{𝑑−2,𝑑−1}. Since 𝑥 is a coregular sequence on 𝐻_𝔪^𝑑−1(𝑀), (0∶_𝐻𝔪^𝑑−1_(𝑀)𝑥) is either zero or co-Cohen-Macaulay of Noetherian dimension 𝑑−3 and so is 𝐻_𝔪^𝑑−2(𝑀/𝑥𝑀). Hence by induction hypothesis 𝐻_𝔪^𝑑−1(𝑀/𝑥𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑−1. Since 𝑥 is a coregular sequence on 𝐻^𝑑_𝔪(𝑀), it follows, by (‡), that 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑. This complete the proof.

The following consequence follows by Theorem 5.

Corollary 6. Let 𝑀 be a Cohen-Macaulay module. Then 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑.

The following theorem extends [16, Corollary 3.6].

Theorem 7. Let width(𝐻^𝑖_𝔪(𝑀))≥𝑖−1 for all 𝑖 with 2≤𝑖<𝑑. Then 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑.

Proof. We use induction on 𝑑. Let 𝑑=3. Then, by [1, Corollary 2.1.7] and our assumption, there exists 𝑥∈𝔪⧵(∪_{𝔭∈Ass(𝑀)}𝔭)∪(∪_{𝔮∈Att(𝐻}²_𝔪(𝑀))𝔮). Hence from the exact sequence 0⟶𝑀^𝑥𝑀⟶𝑀⟶(∗),𝑥𝑀⟶0 (5) we get the exact sequence 0⟶𝐻²_𝔪𝑀𝑥𝑀⟶𝐻³_𝔪(𝑀)⟶^𝑥𝐻³_𝔪(𝑀)⟶0. (6) Therefore 𝐻²_𝔪(𝑀/𝑥𝑀)≅(0∶_𝐻³_𝔪(𝑀)𝑥) and so (0∶_𝐻³_𝔪(𝑀)𝑥) is co-Cohen-Macaulay of Noetherian dimension 2. Thus 𝐻³_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 3. The result has been proved in this case. Now suppose that 𝑑>3 and assume that our assertion is true for 𝑑−1. There exists 𝑥∈𝔪⧵(∪_{𝔭∈Ass(𝑀)}𝔭)∪(∪^𝑑−1_𝑖=2∪_{𝔮∈Att(𝐻}^𝑖_𝔪(𝑀))𝔮) and so from the exact sequence (∗) we have the following long exact sequence ⋯⟶𝐻^𝑖_𝔪𝑀𝑥𝑀⟶𝐻^𝑖_𝔪(𝑀)⟶^𝑥𝐻^𝑖_𝔪(𝑀)⟶⋯. (7) Thus there is an isomorphism 𝐻^𝑖_𝔪(𝑀/𝑥𝑀)≅(0∶_𝐻𝔪^𝑖+1_(𝑀)𝑥)(⋆) for all 𝑖 with 2≤𝑖<𝑑. Hence width(𝐻^𝑖_𝔪(𝑀/𝑥𝑀))≥𝑖−1 for all 𝑖 with 2≤𝑖<𝑑−1 and so by the induction hypothesis 𝐻_𝔪^𝑑−1(𝑀/𝑥𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑−1. Therefore, in view of (⋆), the module 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑, as required.

The following corollary immediately follows by Theorem 7 and [8, Corollary 2.5].

Corollary 8. Let 𝐻^𝑖_𝔪(𝑀) be co-Cohen-Macaulay of Noetherian dimension 𝑖 for all 𝑖 with 0≤𝑖<𝑑. Then 𝐻^𝑑_𝔪(𝑀) is co-Cohen-Macaulay of Noetherian dimension 𝑑.

Acknowledgments

The authors are deeply grateful to the referees for their careful reading of the paper and the helpful suggestions.