# Co-Cohen-Macaulay Modules and Local Cohomology

Authors: Hero Saremi; Amir Mafi
Journal: Journal of Mathematics
 (2013)

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## 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 (H𝔪i(M))≥i-1 for all i with 2≤i<d, then H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d; (b) if M is an unmixed R-module and depth M≥d-1, then H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d if and only if H𝔪d-1(M) is either zero or co-Cohen-Macaulay of Noetherian dimension d-2. As consequence, if H𝔪i(M) is co-Cohen-Macaulay of Noetherian dimension i for all i with 0≤i<d, then H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d.


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## Body

## 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 H𝔪i(M). It is well known that H𝔪i(M) is Artinian for all i (cf. [1]).TheNoetherian 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(An+1/An)<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,…,xn∈𝔪. x1,…,xn is an A-coregular sequence if 0:A(x1,…,xi-1)R→xi0:A(x1,…,xi-1)R is surjective for i=1,…,n and 0:A(x1,…,xn)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 H𝔪d(M) is co-Cohen-Macaulay (see also [9]). Following Nagata [10], M is unmixed if dim(R^/𝔭)=d for all 𝔭∈AssR^M^.The main aim of this paper is to prove the following theorem.Theorem 1.
The following statements are true.(a)
Ifwidth(H𝔪i(M))≥i-1 for all i with 2≤i<d, then H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d.(b)
IfM is an unmixed R-module and depthM≥d-1, then H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d if and only if H𝔪d-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+⋯+An, where Ai is 𝔭i secondary. The set {p1,…,pn} 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,…,xn) of elements in 𝔪 is called a strict f-sequence of M if xj+1∉𝔭 for all 𝔭∈∪i=1d-jAtt(H𝔪i(M/(x1,…,xj)M)∖{𝔪} for all j=0,1,…,n-1. This notion was introduced in [12].Lemma 2 (see [9]).
For all integer0≤i<d, one has N-dim(H𝔪i(M))≤i and N-dim(H𝔪d(M))=d.Lemma 3.
Letx∈𝔪 be a strict f-sequence of M. Then the following statements are true.(i)
Suppose thatd≥3 and H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d. Then H𝔪d-1(M/xM) is also co-Cohen-Macaulay of Noetherian dimension d-1.(ii)
Suppose thatH𝔪d-1(M/xM) is co-Cohen-Macaulay of Noetherian dimension d-1 and width(H𝔪d(M))≥3. Then H𝔪d(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
(1)0⟶(0:Mx)⟶M⟶M0:Mx⟶0,0⟶M(0:Mx)⟶xM⟶MxM⟶0,
we get the exact sequence
(2)0⟶H𝔪i(M)xH𝔪i(M)⟶H𝔪i(MxM)⟶(0:H𝔪i+1(M)x)⟶0(†),
for all i=1,2,…,d-1. Thus in case i=d-1 we have that (0:H𝔪d(M)x) is co-Cohen-Macaulay of Noetherian dimension d-1. By the choice of x∈𝔪 the module H𝔪d-1(M)/xH𝔪d-1(M) is an R-module of finite length. Moreover, since width(H𝔪d-1(M/xM))≥min{2,d-1}>0 by [8, Proposition 2.4], we have H0𝔪(H𝔪d-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 H𝔪d-1(M)/xH𝔪d-1(M)=0. Hence there is an isomorphism H𝔪d-1(M/xM)≅(0:H𝔪d(M)x) and so H𝔪d-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):H𝔭R𝔭i-dimR/𝔭(M𝔭)≠0}, the ith pseudo support of M, and denoted by Psuppi(M). Note that if R is complete with respect to 𝔪-adic topology, then by [15, Theorem 3.1] Var(Ann(H𝔪i(M)))=Psuppi(M). The module M satisfies Serre's condition Sn, where n is nonnegative integer, provided depthM𝔭≥min{n,dimM𝔭} 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.
LetM be unmixed and d≤2. Then H𝔪d-1(M) is either zero or co-Cohen-Macaulay of Noetherian dimension d-2.Proof.
Whend=0, it is trivial. We assume that d=1. Since M is unmixed then depthM≥1 and so we have H𝔪0(M)=0. Thus the result has been proved in this case. Now assume that d=2 and H𝔪1(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 H𝔭R𝔭1-dimR/𝔭(M𝔭)≠0. Therefore 1≤depthM𝔭≤1-dimR/𝔭 and so dimR/𝔭=0. Hence Psupp1(M)={𝔪} and so by [15, Theorem 3.1] Att(H𝔪1(M))⊆{𝔪}. This implies that H𝔪1(M) is of finite length (see [1, Corollary 7.2.12]). Hence H𝔪1(M) is co-Cohen-Macaulay of Noetherian dimension zero.Theorem 5.
LetM be an unmixed R-module. If depthM≥d-1, then the following statements are equivalent: (i)
the moduleH𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d;(ii)
the moduleH𝔪d-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, H𝔪d-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 H𝔪d-2(M/xM) is either zero or co-Cohen-Macaulay of Noetherian dimension d-3. Hence, from the exact sequence
(3)0⟶H𝔪d-2(M)xH𝔪d-2(M)⟶H𝔪d-2(MxM)⟶(0:H𝔪d-1(M)x)⟶0,
and our assumption we get H𝔪d-2(M/xM)≅(0:H𝔪d-1(M)x) and so (0:H𝔪d-1(M)x) is either zero or co-Cohen-Macaulay of Noetherian dimension d-3. Since x is a coregular sequence on H𝔪d-1(M), we have H𝔪d-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(H𝔪d-1(M))𝔮). Hence, from the exact sequence
(4)0⟶H𝔪i(M)xH𝔪i(M)⟶H𝔪i(MxM)⟶(0:H𝔪i+1(M)x)⟶0,
we have the isomorphism H𝔪i(M/xM)≅(0:H𝔪i+1(M)x)(‡) for i∈{d-2,d-1}. Since x is a coregular sequence on H𝔪d-1(M), (0:H𝔪d-1(M)x) is either zero or co-Cohen-Macaulay of Noetherian dimension d-3 and so is H𝔪d-2(M/xM). Hence by induction hypothesis H𝔪d-1(M/xM) is co-Cohen-Macaulay of Noetherian dimension d-1. Since x is a coregular sequence on H𝔪d(M), it follows, by (‡), that H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d. This complete the proof.The following consequence follows by Theorem5.Corollary 6.
LetM be a Cohen-Macaulay module. Then H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d.The following theorem extends [16, Corollary 3.6].Theorem 7.
Letwidth(H𝔪i(M))≥i-1 for all i with 2≤i<d. Then H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d.Proof.
We use induction ond. Let d=3. Then, by [1, Corollary 2.1.7] and our assumption, there exists x∈𝔪∖(∪𝔭∈Ass(M)𝔭)∪(∪𝔮∈Att(H𝔪2(M))𝔮). Hence from the exact sequence
(5)0⟶M⟶xM⟶MxM⟶0(*),
we get the exact sequence
(6)0⟶H𝔪2(MxM)⟶H𝔪3(M)⟶xH𝔪3(M)⟶0.
Therefore H𝔪2(M/xM)≅(0:H𝔪3(M)x) and so (0:H𝔪3(M)x) is co-Cohen-Macaulay of Noetherian dimension 2. Thus H𝔪3(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)∪(∪i=2d-1∪𝔮∈Att(H𝔪i(M))𝔮) and so from the exact sequence (*) we have the following long exact sequence
(7)⋯⟶H𝔪i(MxM)⟶H𝔪i(M)⟶xH𝔪i(M)⟶⋯.
Thus there is an isomorphism H𝔪i(M/xM)≅(0:H𝔪i+1(M)x)(⋆) for all i with 2≤i<d. Hence width(H𝔪i(M/xM))≥i-1 for all i with 2≤i<d-1 and so by the induction hypothesis H𝔪d-1(M/xM) is co-Cohen-Macaulay of Noetherian dimension d-1. Therefore, in view of (⋆), the module H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d, as required.The following corollary immediately follows by Theorem7 and [8, Corollary 2.5].Corollary 8.
LetH𝔪i(M) be co-Cohen-Macaulay of Noetherian dimension i for all i with 0≤i<d. Then H𝔪d(M) is co-Cohen-Macaulay of Noetherian dimension d.



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