Notes on linkage of modules

TL;DR

This paper investigates how the Cohen-Macaulay property is preserved under linkage in local rings and explores the relationship between linked modules' properties and local cohomology.

Contribution

It demonstrates conditions under which Cohen-Macaulayness is preserved under linkage and analyzes the connection between (S_n) loci and local cohomology attached primes.

Findings

Cohen-Macaulayness is preserved under linkage under mild conditions.

The (S_n) locus of linked modules relates to attached primes of local cohomology.

Provides new insights into the structure of linked modules in Cohen-Macaulay rings.

Abstract

Let R be a Cohen-Macaulay local ring. It is shown that under some mild conditions, the Cohen-Macaulayness property is preserved under linkage. We also study the connection of (S_n) locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module.

Equations

$$0⟶M^{\ast }⟶P_0^{\ast }⟶\lambda M⟶0,$$

$$0⟶\lambda M⟶P_1^{\ast }⟶\mathsf{Tr}M⟶0.$$

$$0\to X_n\to X_{n-1}\to \cdots \to X_1\to X_0\to M\to 0,$$

$$0\to M^{▽}\to P_0^{▽}\to P_1^{▽}\to {\mathsf{Tr}}_{C}M\to 0.$$

$${\tilde{S}}_k(M)=\{\mathfrak{p}\in \mathrm{Spec}R\mid M_{\mathfrak{p}}\text{ satisfies }{\tilde{S}}_k\}.$$

$${\mathrm{Att}}_{R_{\mathfrak{p}}}(H_{\mathfrak{p}{R}_{\mathfrak{p}}}^{i-\dim R/\mathfrak{p}}(M_{\mathfrak{p}}))=\{\mathfrak{q}{R}_{\mathfrak{p}}\mid \mathfrak{q}\in {\mathrm{Att}}_R(H_{\mathfrak{m}}^i(M),\mathfrak{q}\subseteq \mathfrak{p})\},$$

$$0\to {\mathrm{Ext}}_R^1(\mathsf{Tr}M,R)\to M\to M^{\ast \ast }\to {\mathrm{Ext}}_R^2(\mathsf{Tr}M,R)\to 0.$$

$$0\to {\mathrm{Ext}}_R^1(\mathsf{Tr}M,C)\to M{\otimes }_{R}C\to {\mathrm{Hom}}_R(M^{\ast },C)\to {\mathrm{Ext}}_R^2(\mathsf{Tr}M,C)\to 0.$$

$$0⟶M⟶F⟶M_1⟶0,$$

$$0⟶\mathsf{Tr}{M}_1⟶P⟶\mathsf{Tr}M⟶0,$$

$$\begin{array}[]{rl}\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{n+1}_{R}(\mathsf{Tr}M,R))&=\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{n}_{R}(\mathsf{Tr}M_{1},R))\\ &=\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{n}_{R}(\mathsf{Tr}M_{1},C))=\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{n+1}_{R}(\mathsf{Tr}M,C)).\end{array}$$

$$t=\min \{j\mid {\mathrm{Ext}}_R^j(\mathsf{Tr}M,R{)}_{{\mathfrak{p}}_0}\ne 0,1\le j\le n\}.$$

$${\mathrm{Ext}}_R^t(\mathsf{Tr}M,R{)}_{\mathfrak{q}}=0\text{ for all prime ideal }\mathfrak{q}\text{ with }\mathfrak{q}⊊{\mathfrak{p}}_0.$$

$${\mathrm{Ass}}_R({\mathrm{Ext}}_R^t(\mathsf{Tr}M,R))=\{P\cap R\mid P\in {\mathrm{Ass}}_{\hat{R}}({\mathrm{Ext}}_{\hat{R}}^t(\widehat{\mathsf{Tr}M},\hat{R}))\}.$$

$${\mathrm{Ext}}_{\hat{R}}^i(\widehat{\mathsf{Tr}M},\hat{R}{)}_{P_0}=0\text{ for all }0<i<t.$$

$$P_0{\hat{R}}_{P_0}\in \mathrm{Ass}({\mathrm{Ext}}_{{\hat{R}}_{P_0}}^t({\mathsf{Tr}}_{{\hat{R}}_{P_0}}{\hat{M}}_{P_0},{\omega }_{{\hat{R}}_{P_0}})).$$

$$H_{\mathfrak{p}{R}_{\mathfrak{p}}}^{i-\dim R/\mathfrak{p}}(M_{\mathfrak{p}})\ne 0.$$

$${\mathrm{depth}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\ge \min \{\dim R_{\mathfrak{p}},n\}.$$

$$\dim R_{\mathfrak{p}}+\dim R/\mathfrak{p}=d.$$

$${\mathrm{Att}}_R(H_{\mathfrak{m}}^c(\lambda M))=\left\{\mathfrak{p}\in S_{d-c}(M)\setminus S_{d-c+1}(M)\mid {\mathrm{depth}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})=d-c\right\}.$$

$${\mathrm{Att}}_R(H_{\mathfrak{m}}^c(\lambda M))=\left\{P\cap R\mid P\in {\mathrm{Att}}_{\hat{R}}(H_{\mathfrak{m}}^c(\lambda M))\right\}.$$

$${\mathrm{Att}}_{\hat{R}}(H_{\mathfrak{m}}^c(\lambda M))={\mathrm{Ass}}_{\hat{R}}({\mathrm{Ext}}_{\hat{R}}^{d-c}(\widehat{(\lambda M)},{\omega }_{\hat{R}}))={\mathrm{Ass}}_{\hat{R}}({\mathrm{Ext}}_{\hat{R}}^{d-c}(\widehat{(\lambda M)},\hat{R})).$$

$${\mathrm{Ass}}_R({\mathrm{Ext}}_R^{d-c}((\lambda M),R))=\left\{P\cap R\mid P\in {\mathrm{Ass}}_{\hat{R}}({\mathrm{Ext}}_{\hat{R}}^{d-c}(\widehat{(\lambda M)},\hat{R}))\right\}.$$

$${\mathrm{Att}}_R(H_{\mathfrak{m}}^c(\lambda M))={\mathrm{Ass}}_R({\mathrm{Ext}}_R^{d-c}((\lambda M),R)).$$

$$0⟶M⟶{\Omega }^{n+1}\mathsf{Tr}{\Omega }^{n+1}\mathsf{Tr}M\oplus P⟶{\Omega }^{n-1}{\mathrm{Ext}}_R^{n+1}(\mathsf{Tr}M,R)⟶0,$$

$${\mathrm{Att}}_R(H_{\mathfrak{m}}^{d-n}(\lambda M))\setminus \overline{\bigcup _{d-n<i<d}{\mathrm{Att}}_R(H_{\mathfrak{m}}^i(\lambda M))}=\left\{\mathfrak{p}\in S_n(M)\setminus S_{n+1}(M)\mid {\mathrm{depth}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})=n\right\}.$$

$$\mathfrak{p}\in S_n(M)⟺H_{\mathfrak{p}{R}_{\mathfrak{p}}}^i((\lambda M{)}_{\mathfrak{p}})=0\text{ for }\dim R_{\mathfrak{p}}-n<i<\dim R_{\mathfrak{p}}.$$

$$0\to {\mathrm{Hom}}_R(M,\mathfrak{c})\to {\mathrm{Hom}}_R(F_0,\mathfrak{c})\to {\mathrm{Hom}}_R(F_1,\mathfrak{c})\to {\mathsf{Tr}}_{\mathfrak{c}}(M)\to 0.$$

$$\text{setcounter}MaxMatrixCols14\begin{array}{ccccccccc}& & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ \to & \ker \varphi & \stackrel{\iota }{\to }& (F_0{)}^{\ast }{\otimes }_R\mathfrak{c}& \stackrel{\varphi }{\to }& \mathfrak{c}\lambda M& \to & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ \downarrow \cong & & \downarrow \cong & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ 0& \to & {\mathrm{Hom}}_R(M,\mathfrak{c})& \to & {\mathrm{Hom}}_R(F_0,\mathfrak{c})& \to & {\lambda }_{\mathfrak{c}}(M)& \to & 0\end{array}$$

$${\lambda }_{\mathfrak{c}}(M)\cong \mathfrak{c}\lambda M.$$

Full text

This research was supported by a grant from IPM

1. introduction

The theory of linkage for subschemes of projective space goes back more than a century in some sense, but the modern study was introduced by Peskine and Szpiro [27] in 1974. Recall that two ideals $I$ and $J$ in a Cohen-Macaulay local ring $R$ are said to be linked if there is a regular sequence $\alpha$ in their intersection such that $I=(\alpha:J)$ and $J=(\alpha:I)$. The first main result in the theory of linkage, due to Peskine and Szpiro, indicates that the Cohen-Macaulayness property is preserved under linkage over Gorenstein rings. They also give a counterexample to show that the above result is no longer true if the base ring is Cohen-Macaulay but not Gorenstein. On the other hand, there are interesting extensions of the Peskine-Szpiro Theorem to the case that $R$ is Cohen-Macaulay and one of the ideals $I$ or $J$ is strongly Cohen-Macaulay (i.e. Koszul homology modules are Cohen-Macaulay) [15] or satisfies the sliding depth condition (on Koszul homology) [14]. In a different direction, Schenzel generalized the Peskine-Szpiro Theorem to the vanishing of certain local cohomology modules. More precisely, for linked ideals $I$ and $J$ over a Gorenstein local ring $R$ with maximal ideal $\mathfrak{m}$, the Serre condition $(S_{n})$ for $R/I$ is equivalent to the vanishing of the local cohomology modules $\operatorname{H}^{i}_{\mathfrak{m}}(R/J)=0$ for all $i$, $\dim(R/J)-n<i<\dim(R/J)$ [32, Theorem 4.1].

A few authors advanced the theory to the setting of linkage of modules in different ways, for instance Martin [19], Yoshino and Isogawa [35], Martsinkovsky and Strooker [20], and Nagel [24]. Based on these generalizations, several works have been done on studying the linkage theory in the context of modules; see for example [6]–[10], [16], [26], [28] and [29]. In this paper, we are interested in linkage of modules in the sense of [20]. Martsinkovsky and Strooker generalized the notion of linkage for modules by using the composition of two functors: transpose and syzygy. They showed that ideals $\mathfrak{a}$ and $\mathfrak{b}$ are linked by zero ideal if and only if $R/\mathfrak{a}\cong\lambda(R/\mathfrak{b})$ and $R/\mathfrak{b}\cong\lambda(R/\mathfrak{a})$, where $\lambda:=\Omega\mathsf{Tr}$.

Recall that the $(S_{k})$ locus of $M$, denoted by $S_{k}(M)$, is the subset of $\operatorname{Spec}R$ consisting of all prime ideals $\mathfrak{p}$ of $R$ such that $M_{\mathfrak{p}}$ satisfies the Serre condition $(S_{k})$. The Gorenstein locus of $R$, denoted by $\operatorname{Gor}(R)$, is the subset of $\operatorname{Spec}R$ consisting of all prime ideals $\mathfrak{p}$ of $R$ such that $R_{\mathfrak{p}}$ is a Gorenstein local ring. In the fisrt part of this paper, we study the connection of $(S_{n})$ locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module. As a consequence, we obtain the following result (see also Theorem 3.3 for a more general case).

Theorem A. Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and let $X$ be an open subset of $\operatorname{Spec}R$ which is contained in $\operatorname{Gor}(R)$. Assume that $M$ is a horizontally linked $R$–module and that $n$ is a positive integer. Then $X\subseteq S_{n}(M)$ if and only if $\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(\lambda M))\subseteq\operatorname{Spec}R\setminus X$ for $d-n<i<d$.

Note that for an Artinian $R$–module $N$, $\operatorname{Att}_{R}(N)=\emptyset$ if and only if $N=0$. Therefore, the above theorem can be viewed as a generalization of Schenzel’s result. We should note that Theorem A is new even in the case of cyclic module. The above theorem helps us to study the attached primes of local cohomology modules of a linked module. As an application of Theorem A, for a horizontally linked module $M$ over a Gorenstein local ring $R$ and a closed subset $X$ of $\operatorname{Spec}R$, it is shown that $\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(M))\subseteq X$ for $0<i<\dim R$ if and only if $\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(\lambda M))\subseteq X$ for $0<i<\dim R$ (see Corollary 3.8).

There is another interesting generalization of the Peskine-Szpiro Theorem, due to Martin [18]. Let $R$ be a Gorenstein local ring and let $I$ be a Cohen-Macaulay ideal linked to $J$ by an ideal $\mathfrak{a}$, where $\mathfrak{a}$ is Cohen-Macaulay and generically Gorenstein. Then $J$ is Cohen-Macaulay if and only if $I+\mathfrak{b}$ is a Cohen-Macaulay ideal with the same height as $\mathfrak{b}$, where $\mathfrak{b}$ is an ideal of $R$, containing $\mathfrak{a}$, with $\mathfrak{b}/\mathfrak{a}$ equal to a canonical module of $\omega_{R/\mathfrak{a}}$ [18, Corolary 2.11]. In the second part of this paper, we study the theory of linkage for Cohen-Macaulay modules. As a consequence, we generalize the Martin’s result to the module case. More precisely, we have the following result (see also Corollary 4.3 for a more general case).

Theorem B. *Let $R$ be a Cohen-Macaulay local ring with canonical module $\omega_{R}$ and let $\mathfrak{a}$ be a generically Gorenstein, Cohen-Macaulay ideal of $R$. Write $\omega_{R/\mathfrak{a}}\cong\mathfrak{c}/\mathfrak{a}$. Assume that $M$ is a Cohen-Macaulay $R$–module which is linked to an $R$–module $N$ by the ideal $\mathfrak{a}$. Then $N$ is Cohen-Macaulay if and only if $\operatorname{\mathsf{depth}}_{R}(M/\mathfrak{c}M)\geq\dim(R/\mathfrak{a})-1$.

The organization of the paper is as follows. In Section 2, we collect preliminary notions, definitions and some known results which will be used in this paper. In Section 3, we study the connection of $(S_{n})$ locus of a horizontally linked module and the attached primes of certain local cohomology modules of its linked module. As a consequence, we prove Theorem A in this section. In section 4, we study the theory of linkage for Cohen-Macaulay modules. We prove Theorem B in this section. In section 5, we establish a duality on local cohomology modules of a linked module.

2. Definitions and preliminary results

Throughout the paper $R$ is a commutative Noetherian local ring and all modules over $R$ are assumed to be finitely generated. We start by recalling several definitions and terminology from [2, 5, 20].

** 2.1**** ([2]).**

Let $M$ be an $R$–module. For a positive integer $i$, we denote by $\Omega^{i}M$ the $i$-th syzygy of M, namely, the image of the $i$-th differential map in a minimal free resolution of $M$.

The transpose $\mathsf{Tr}M$ of $M$ is defined as the cokernel of the $R$-dual map $\partial_{1}^{\ast}=\operatorname{\mathsf{Hom}}_{R}(\partial_{1},R)$ of the first differential map $\partial_{1}$ in a minimal free resolution of $M$. Hence there is an exact sequence of the form $0\rightarrow M^{*}\rightarrow P_{0}^{*}\rightarrow P_{1}^{*}\rightarrow\mathsf{Tr}M\rightarrow 0$. Note that the modules $\Omega^{i}M$ and $\mathsf{Tr}M$ are uniquely determined up to isomorphism, since so is a minimal free resolution of $M$, and there is a stable isomorphism $\Omega^{2}\mathsf{Tr}M\cong M^{\ast}$.

A stable module is a module with no non-zero free direct summands. An $R$–module $M$ is called a syzygy module if it is embedded in a free $R$–module.

** 2.2****.**

Linkage of modules.([20]) Two $R$–modules $M$ and $N$ are said to be horizontally linked if $M\cong\lambda N$ and $N\cong\lambda M$, where $\lambda M:=\Omega\mathsf{Tr}M$. Equivalently, $M$ is horizontally linked (to $\lambda M$) if and only if $M\cong\lambda^{2}M$. An $R$–module $M$ is horizontally linked if and only if it is stable and $\operatorname{\mathsf{Ext}}^{1}_{R}(\mathsf{Tr}M,R)=0$, equivalently $M$ is stable and is a syzygy module [20, Theorem 2 and Proposition 3]. An $R$–module $M$ is said to be linked to an $R$–module $N$, by an ideal $\mathfrak{c}$ of $R$, if $\mathfrak{c}\subseteq\operatorname{ann}_{R}(M)\cap\operatorname{ann}_{R}(N)$ and $M$ and $N$ are horizontally linked as $R/{\mathfrak{c}}$–modules. In this situation we write $M\underset{\mathfrak{c}}{\sim}N$.

Let $M$ be a stable $R$–module and let $P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0$ be a minimal projective presentation of $M$. Then $P_{0}^{*}\rightarrow P_{1}^{*}\rightarrow\mathsf{Tr}M\rightarrow 0$ is a minimal projective presentation of $\mathsf{Tr}M$ [1, Theorem 32.13]. We quote the following induced exact sequences

[TABLE]

[TABLE]

** 2.3****.**

Gorenstein dimension with respect to a semidualizing module. ([11], [13]) An $R$–module $C$ is called a semidualizing module, if the homothety morphism $R\to\operatorname{\mathsf{Hom}}_{R}(C,C)$ is an isomorphism and $\operatorname{\mathsf{Ext}}^{i}_{R}(C,C)=0$ for all $i>0$. Semidualizing modules are initially studied by Foxby [11] and Golod [13]. It is clear that $R$ itself is a semidualizing $R$–module. Over a Cohen-Macaulay local ring $R$, a canonical module $\omega_{R}$ of $R$ is a semidualizing module with finite injective dimension.

An $R$–module $M$ is said to be totally $C$-reflexive if the natural map $M\to\operatorname{\mathsf{Hom}}(\operatorname{\mathsf{Hom}}(M,C),C)$ is bijective and $\operatorname{\mathsf{Ext}}_{R}^{i}(M,C)=0=\operatorname{\mathsf{Ext}}_{R}^{i}(\operatorname{\mathsf{Hom}}(M,C),C)$ for all $i\geq 1$. The smallest nonnegative integer $n$ for which there exists an exact sequence

[TABLE]

such that each $X_{i}$ is totally $C$-reflexive, is called the $\operatorname{\mathsf{G}_{\it C}}$-dimension of $M$. If $M$ has $\operatorname{\mathsf{G}_{\it C}}$-dimension $n$, we write $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(M)=n$. Therefore $M$ is totally $C$-reflexive if and only if $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(M)=0$.

Let $P_{1}\overset{f}{\rightarrow}P_{0}\rightarrow M\rightarrow 0$ be a projective presentation of $M$. The transpose of $M$ with respect to $C$, $\mathsf{Tr}_{C}M$, is defined to be $\operatorname{Coker}f^{\triangledown}$, where $(-)^{\triangledown}:=\operatorname{\mathsf{Hom}}_{R}(-,C)$, which satisfies the exact sequence

[TABLE]

We catalogue some properties of Gorenstein dimension with respect to a semidualizing module:

1. (i)

If $C$ is a dualizing $R$–module, then $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(M)<\infty$ for all $R$–modules $M$; 2. (ii)

If $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(M)<\infty$, then $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(M)=\operatorname{\mathsf{depth}}(R)-\operatorname{\mathsf{depth}}(M)$; 3. (iii)

If $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(M)<\infty$, then $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(M)=\sup\{i\mid\operatorname{\mathsf{Ext}}^{i}_{R}(M,C)\neq 0\};$ 4. (iv)

$\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(M)=0$* if and only if $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(\mathsf{Tr}_{C}M)=0$.*

An $R$–module $M$ is said to satisfy the property $\widetilde{S}_{k}$ if $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq\min\{k,\operatorname{\mathsf{depth}}R_{\mathfrak{p}}\}$ for all $\mathfrak{p}\in\operatorname{Spec}R$. Recall that the $\widetilde{S}_{k}$ locus of $M$, denoted by $\widetilde{S}_{k}(M)$, is defined by

[TABLE]

For a horizontally linked module $M$ over a Cohen-Macaulay local ring $R$, the properties $\widetilde{S}_{k}$ and $(S_{k})$ are identical. An $R$–module $M$ is called $n$-torsionfree, when $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}M,R)=0$ for $1\leq i\leq n$.

The following Theorem is due to Auslander and Bridger.

Theorem 2.4**.**

[2, Theorem 4.25]** Let $M$ be an $R$–module of finite Gorenstein dimension. Then $M$ is $n$-torsionfree if and only if $M$ satisfies $\widetilde{S}_{n}$.

** 2.5****.**

Auslander class. Let $C$ be a semidualizing $R$-module. The Auslander class with respect to $C$, denoted by $\mathcal{A}_{C}$, consists of all $R$–modules $M$ satisfying the following conditions.

1. (i)

The natural map $\mu:M\to\operatorname{\mathsf{Hom}}_{R}(C,M\otimes_{R}C)$ is an isomorphism. 2. (ii)

$\operatorname{\mathsf{Tor}}^{R}_{i}(M,C)=0=\operatorname{\mathsf{Ext}}^{i}_{R}(C,M\otimes_{R}C)$* for all $i>0$.*

Every module of finite projective dimension is in the Auslander class $\mathcal{A}_{C}$. Over a Cohen-Macaulay local ring $R$ with canonical module $\omega_{R}$, we have $M\in\mathcal{A}_{\omega_{R}}$ if and only if $\operatorname{\mathsf{G-dim}}_{R}(M)<\infty$ [12, Theorem 1].

** 2.6****.**

Secondary representation.([17]) Let $M$ be a nonzero Artinian $R$–module. We say that $M$ is secondary if the multiplication by $x$ on $M$ is surjective or nilpotent for every $x\in R$. In this case, $\mathfrak{p}:=\sqrt{\operatorname{ann}_{R}(M)}$ is a prime ideal of $R$ and we say that $M$ is $\mathfrak{p}$-secondary. Note that every Artinian $R$–module $M$ has a minimal secondary representation $M=M_{1}+\cdots+M_{n}$, where $M_{i}$ is $\mathfrak{p}_{i}$-secondary, each $M_{i}$ is not redundant and $\mathfrak{p}_{i}\neq\mathfrak{p}_{j}$ for all $i\neq j$. The set $\{\mathfrak{p}_{1},\cdots,\mathfrak{p}_{n}\}$ is independent of the choice of the minimal secondary representation of $M$. This set is called the set of attached primes of $M$ and denoted by $\operatorname{Att}_{R}(M)$.

We denote by $\widehat{M}$ the completion of $M$ in the $\mathfrak{m}$-adic topology. In the following we collect some basic properties of attached primes of a module which will be used throughout the paper (see [17] and [5]).

Theorem 2.7**.**

Let $M$ be an Artinian $R$–module. The following statements hold.

1. (i)

$M\neq 0$* if and if $\operatorname{Att}_{R}(M)\neq\emptyset.$* 2. (ii)

$M$* has finite length if and only if $\operatorname{Att}_{R}(M)\subseteq\{\mathfrak{m}\}$.* 3. (iii)

$\operatorname{Att}_{R}(M)=\operatorname{Ass}_{R}(\operatorname{\mathsf{Hom}}_{R}(M,\operatorname{E}_{R}(k)))$, where $\operatorname{E}_{R}(k)$ is the injective envelope of $k$. 4. (iv)

$\operatorname{Att}_{R}(M)=\{P\cap R\mid P\in\operatorname{Att}_{\widehat{R}}(M)\}$.

The following result plays an important role in the proof of Theorem A.

Theorem 2.8**.**

[25, Theorem 1.1]** Let $R$ be a ring which is the homomorphic image of a Cohen-Macaulay local ring. Then

[TABLE]

for every $R$–module $M$, integer $i\geq 0$ and prime ideal $\mathfrak{p}\in R$.

3. Linkage and the attached primes of local cohomology modules

Recall that a subset $X$ of $\operatorname{Spec}R$ is called stable under generalization provided that if $\mathfrak{p}\in X$ and $\mathfrak{q}\in\operatorname{Spec}R$ with $\mathfrak{q}\subseteq\mathfrak{p}$, then $\mathfrak{q}\in X$. Note that every open subset of $\operatorname{Spec}R$ is stable under generalization. For an integer $n$, set $X^{n}(R):=\{\mathfrak{p}\in\operatorname{Spec}(R)\mid\operatorname{\mathsf{depth}}R_{\mathfrak{p}}\leq n\}$.

We start by recalling the following result.

Lemma 3.1**.**

Let $C$ be a semidualizing $R$–module and let $M$ be an $R$–module. Assume that $M\in\mathcal{A}_{C}$ and that $n$ is a positive integer. Consider the following statements.

1. (i)

$\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}M,R)=0$* for $1\leq i\leq n$.* 2. (ii)

$\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}M,C)=0$* for $1\leq i\leq n$.* 3. (iii)

$M$* satisfies $\widetilde{S}_{n}$.*

Then we have the following.

1. (a)

(i)$\Leftrightarrow$(ii)$\Rightarrow$(iii) 2. (b)

If $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$ for all $\mathfrak{p}\in\operatorname{X}^{n-1}(R)$ (e.g. $\operatorname{id}_{R_{\mathfrak{p}}}(C_{\mathfrak{p}})<\infty$ for all $\mathfrak{p}\in\operatorname{X}^{n-1}(R)$), then all the statements (i)-(iii) are equivalent.

Proof.

See [31, Theorem 4.1] and [9, Theorem 2.12]. ∎

The proof of Theorem A is based on the following lemma, which is of independent interest.

Lemma 3.2**.**

Let $C$ be a semidualizing $R$–module and let $M$ be an $n$-torsionfree $R$–module. Assume that $M\in\mathcal{A}_{C}$ and that $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$ for all $\mathfrak{p}\in X^{n}(R)$ (e.g. $\operatorname{id}_{R_{\mathfrak{p}}}(C_{\mathfrak{p}})<\infty$ for all $\mathfrak{p}\in X^{n}(R)$). Then $\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{n+1}_{R}(\mathsf{Tr}M,C))=\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{n+1}_{R}(\mathsf{Tr}M,R)).$

Proof.

We argue by induction on $n$. Let $n=0$ and let $\mathfrak{p}\in\operatorname{Ass}(\operatorname{\mathsf{Ext}}^{1}_{R}(\mathsf{Tr}M,R))$. Consider the following exact sequence (see [21, Proposition 5])

[TABLE]

It follows from the exact sequence (3.2.1) that $\mathfrak{p}\in\operatorname{Ass}_{R}(M)$. As $M\in\mathcal{A}_{C}$, we have $M\cong\operatorname{\mathsf{Hom}}_{R}(C,C\otimes_{R}M)$. Therefore $\mathfrak{p}\in\operatorname{Ass}_{R}(M\otimes_{R}C)$. Let us now consider the following exact sequence (see [2, Proposition 2.6])

[TABLE]

If $\operatorname{\mathsf{depth}}R_{\mathfrak{p}}=0$ then $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})=0$ and so $\operatorname{\mathsf{Ext}}^{1}_{R}(\mathsf{Tr}M,R)_{\mathfrak{p}}=0$ which is a contradiction. Therefore, $\operatorname{\mathsf{depth}}R_{\mathfrak{p}}>0$ and so $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(\operatorname{\mathsf{Hom}}_{R}(M^{*},C)_{\mathfrak{p}})\geq\min\{2,\operatorname{\mathsf{depth}}R_{\mathfrak{p}}\}>0$. Hence, it follows from the exact sequence (3.2.2) that $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(\operatorname{\mathsf{Ext}}^{1}_{R}(\mathsf{Tr}M,C)_{\mathfrak{p}})=0$. In other words, $\mathfrak{p}\in\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{1}_{R}(\mathsf{Tr}M,C))$. Similarly, one can prove that $\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{1}_{R}(\mathsf{Tr}M,C))\subseteq\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{1}_{R}(\mathsf{Tr}M,R))$. Now let $n>0$. As $M$ is $n$-torsionfree, $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}M,R)=0=\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}M,C)$ for $1\leq i\leq n$ by Lemma 3.1. Consider the universal pushforward of $M$:

[TABLE]

where $F$ is free and $\operatorname{\mathsf{Ext}}^{1}_{R}(M_{1},R)=0$ (see [21]). Hence, by [2, Lemma 3.9], the exact sequence (3.2.3) induces the following exact sequence

[TABLE]

where $P$ is a free $R$-module. It follows from the exact sequence (3.2.4) that $M_{1}$ is $(n-1)$-torsionfree. Also, by the exact sequence (3.2.3), $M_{1}\in\mathcal{A}_{C}$. Hence, by induction hypothesis

[TABLE]

∎

Now we can present our first main result.

Theorem 3.3**.**

Let $(R,\mathfrak{m})$ be a formally equidimensional local ring of dimension $d$ which is the homomorphic image of a Cohen-Macaulay ring. Assume that $M$ is a horizontally linked $R$–module and that $n$ is a positive integer. Assume further that $X$ is a subset of $\operatorname{Spec}R$ which is stable under generalization and that $R_{\mathfrak{p}}$ is Cohen-Macaulay and $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$ for all $\mathfrak{p}\in X$ (e.g. $X\subseteq\operatorname{Gor}(R)$). Then the following are equivalent.

1. (i)

$X\subseteq\widetilde{S}_{n}(M)$. 2. (ii)

$\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(\lambda M))\subseteq\operatorname{Spec}R\setminus X$* for $d-n<i<d$.*

Proof.

(i)$\Rightarrow$(ii). Assume contrarily that $\mathfrak{p}\in\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(\lambda M))\cap X$ for some $d-n<i<d$. Therefore, $\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Att}_{R_{\mathfrak{p}}}(\operatorname{H}^{i-\dim R/\mathfrak{p}}_{\mathfrak{p}R_{\mathfrak{p}}}(\lambda_{R_{\mathfrak{p}}}M_{\mathfrak{p}}))$ by Theorem 2.8. As $\mathfrak{p}\in\widetilde{S}_{n}(M)\cap X$, we have $\operatorname{H}^{j}_{\mathfrak{p}R_{\mathfrak{p}}}(\lambda_{R_{\mathfrak{p}}}M_{\mathfrak{p}})=0$ for $\dim R_{\mathfrak{p}}-n<j<\dim R_{\mathfrak{p}}$ by [8, Theorem 4.2]. Note that by [23, Theorem 31.5], $R$ is equidimensional and catenary. Hence $d=\dim R_{\mathfrak{p}}+\dim R/\mathfrak{p}$ and so $\dim R_{\mathfrak{p}}-n<i-\dim R/\mathfrak{p}<\dim R_{\mathfrak{p}}$. In particular, $\operatorname{H}^{i-\dim R/\mathfrak{p}}_{\mathfrak{p}R_{\mathfrak{p}}}(\lambda_{R_{\mathfrak{p}}}M_{\mathfrak{p}})=0$ which is a contradiction by Theorem 2.7(i).

(ii)$\Rightarrow$(i). Assume contrarily that $X\nsubseteq\widetilde{S}_{n}(M)$. Set $Y=\{\mathfrak{p}\in X\mid M_{\mathfrak{p}}\text{ does not satisfy }\widetilde{S}_{n}\}$. Let $\mathfrak{p}_{0}\in\mathsf{Min}(Y)$. As $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}_{0}}}(M_{\mathfrak{p}_{0}})<\infty$, $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}M,R)_{\mathfrak{p}_{0}}\neq 0$ for some $1\leq i\leq n$ by Theorem 2.4. Set

[TABLE]

As $M$ is horizontally linked, $\operatorname{\mathsf{Ext}}^{1}_{R}(\mathsf{Tr}M,R)=0$ and so $1<t\leq n$. Note that $\mathfrak{q}\in\widetilde{S}_{n}(M)$ for all prime ideal $\mathfrak{q}$ with $\mathfrak{q}\subsetneq\mathfrak{p}_{0}$, because $\mathfrak{p}_{0}\in\mathsf{Min}(Y)$ and $X$ is stable under generalization. In other words, by Theorem 2.4,

[TABLE]

Therefore $\mathfrak{p}_{0}$ is a minimal element of $\operatorname{Supp}_{R}(\operatorname{\mathsf{Ext}}^{t}_{R}(\mathsf{Tr}M,R))$ and so $\mathfrak{p}_{0}\in\mathsf{Min}_{R}(\operatorname{\mathsf{Ext}}^{t}_{R}(\mathsf{Tr}M,R))$. Note that by [22, Proposition 9.A]

[TABLE]

Let $P_{0}\in\mathsf{Min}_{\widehat{R}}(\operatorname{\mathsf{Ext}}^{t}_{\widehat{R}}(\widehat{\mathsf{Tr}M},\widehat{R}))$ such that $P_{0}\cap R=\mathfrak{p}_{0}$. It follows easily from (3.3.1) that

[TABLE]

As $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}_{0}}}(M_{\mathfrak{p}_{0}})<\infty$ and the local homomorphism $R_{\mathfrak{p}_{0}}\rightarrow\widehat{R}_{P_{0}}$ is flat, $\operatorname{\mathsf{G-dim}}_{\widehat{R}_{P_{0}}}(\widehat{M}_{P_{0}})<\infty$. As $R$ is the homomorphic image of a Cohen-Macaulay ring, all formal fibre of $R$ is Cohen-Macaulay. Therefore, $\widehat{R}_{P_{0}}/{\mathfrak{p}_{0}\widehat{R}_{P_{0}}}\cong(R_{\mathfrak{p}_{0}}/{\mathfrak{p}_{0}R_{\mathfrak{p}_{0}}}\otimes_{R}\widehat{R})_{P_{0}}$ is Cohen-Macaulay and so $\widehat{R}_{P_{0}}$ is Cohen-Macaulay (see [23, Page 181]). Therefore, by Lemma 3.2, (3.3.3) and (2.5),

[TABLE]

As $t>1$, $\operatorname{\mathsf{Ext}}^{t}_{\widehat{R}_{P_{0}}}(\mathsf{Tr}_{\widehat{R}_{P_{0}}}\widehat{M}_{P_{0}},\omega_{\widehat{R}_{P_{0}}})\cong\operatorname{\mathsf{Ext}}^{t-1}_{\widehat{R}_{P_{0}}}(\widehat{\lambda M}_{P_{0}},\omega_{\widehat{R}_{P_{0}}})$. Therefore, by (3.3.5), Theorem 2.7(iii) and the local duality theorem, $P_{0}\widehat{R}_{P_{0}}\in\operatorname{Att}_{\widehat{R}_{P_{0}}}(\operatorname{H}^{s-t+1}_{P_{0}\widehat{R}_{P_{0}}}(\widehat{\lambda M}_{P_{0}}))$ where $s=\dim\widehat{R}_{P_{0}}$. As $\widehat{R}$ is equidimensional, $s+\dim(\widehat{R}/{P_{0}})=d$. Hence, by Theorem 2.8, $P_{0}\in\operatorname{Att}_{\widehat{R}}(\operatorname{H}^{d-t+1}_{\widehat{\mathfrak{m}}}(\widehat{\lambda M}))$ and so $\mathfrak{p}_{0}\in\operatorname{Att}_{R}(\operatorname{H}^{d-t+1}_{m}(\lambda M))$ by Theorem 2.7(iv), which is a contradiction because $d-n<d-t+1<d$ and $\mathfrak{p}_{0}\in X$. ∎

It should be mentioned that Theorem 3.3 is new even in the case of cyclic module.

Corollary 3.4**.**

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and let $X$ be a subset of $\operatorname{Spec}R$ which is stable under generalization. Assume that $I$ and $J$ are ideals of $R$ which are linked by the zero ideal. If $\operatorname{\mathsf{pd}}_{R}(R/I)<\infty$ or $R$ is Gorenstein, then the following are equivalent:

1. (i)

$X\subseteq S_{n}(R/I)$. 2. (ii)

$\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(R/J))\subseteq\operatorname{Spec}R\setminus X$* for $d-n<i<d$.*

As a consequence of Theorem 3.3, we have the following result.

Corollary 3.5**.**

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$. Assume that $R_{\mathfrak{p}}$ is Gorenstein for all $\mathfrak{p}\in X^{n-1}(R)$. For a horizontally linked $R$–module $M$, the following are equivalent:

1. (i)

$M_{\mathfrak{p}}$* is Cohen-Macaulay for all $\mathfrak{p}\in X^{n-1}(R)$.* 2. (ii)

$\operatorname{\mathsf{depth}}R_{\mathfrak{p}}\geq n$* for all $\mathfrak{p}\in\underset{\tiny{0<i<d}}{\bigcup}\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(\lambda M))$.*

Proof.

The condition (i) is equivalent to say that $X^{n-1}(R)\subseteq S_{d}(M)$. Now the assertion is clear by Theorem 3.3. ∎

The following is an immediate consequence of Theorem 3.3 and Theorem 2.7(ii).

Corollary 3.6**.**

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$. Assume that $M$ is a horizontally linked $R$–module and that $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$ for all $\mathfrak{p}\in\operatorname{Spec}R\setminus\{\mathfrak{m}\}$. The following statements are equivalent:

1. (i)

$M_{\mathfrak{p}}$* satisfies $(S_{n})$ for all $\mathfrak{p}\in\operatorname{Spec}R\setminus\{\mathfrak{m}\}$.* 2. (ii)

$\ell(\operatorname{H}^{i}_{\mathfrak{m}}(\lambda M)))<\infty$* for $d-n<i<d$.*

where $\ell(-)$ denotes the length.

Theorem 3.7**.**

Let $(R,\mathfrak{m})$ be a formally equidimensional local ring of dimension $d$ which is the homomorphic image of a Cohen-Macaulay ring. Assume that $M$ is a horizontally linked $R$–module and that $n$ is an integer with $0<n\leq d$. Let $X$ be a subset of $\operatorname{Spec}R$ which is stable under generalization such that $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$ and $R_{\mathfrak{p}}$ is Cohen-Macaulay for all $\mathfrak{p}\in X$. If $\operatorname{Att}_{R}(\operatorname{H}^{j}_{\mathfrak{m}}(\lambda M))\subseteq\operatorname{Spec}R\setminus X$ for $d-n<j<d$, then $\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(M))\subseteq\operatorname{Spec}R\setminus X$ for $0<i<n$.

Proof.

Assume contrarily that $\mathfrak{p}\in\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(M))\cap X$ for some $0<i<n$. By Theorem 2.8, $\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Att}_{R_{\mathfrak{p}}}(\operatorname{H}^{i-\dim R/\mathfrak{p}}_{\mathfrak{p}R_{\mathfrak{p}}}(M_{\mathfrak{p}}))$. In particular,

[TABLE]

By Theorem 3.3, $M_{\mathfrak{p}}$ satisfies $\widetilde{S}_{n}$. In other words, as $R_{\mathfrak{p}}$ is Cohen-Macaulay,

[TABLE]

On the other hand, as $R$ is equidimensional and catenary,

[TABLE]

If $n\geq\dim R_{\mathfrak{p}}$, then $M_{\mathfrak{p}}$ is maximal Cohen-Macaulay by (3.7.2) and so $\operatorname{H}^{j}_{\mathfrak{p}R_{\mathfrak{p}}}(M_{\mathfrak{p}})=0$ for all $j\neq\dim R_{\mathfrak{p}}$. Therefore, by (3.7.1), $i-\dim R/\mathfrak{p}=\dim M_{\mathfrak{p}}=\dim R_{\mathfrak{p}}$ and so $i=d$ by (3.7.3) which is a contradiction. On the other hand, if $n<\dim R_{\mathfrak{p}}$, then $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq n$ by (3.7.2) and so $\operatorname{H}^{j}_{\mathfrak{p}R_{\mathfrak{p}}}(M_{\mathfrak{p}})=0$ for all $j<n$. In particular, $\operatorname{H}^{i-\dim R/\mathfrak{p}}_{\mathfrak{p}R_{\mathfrak{p}}}(M_{\mathfrak{p}})=0$ which is a contradiction by (3.7.1). ∎

Recall that a subset $X$ of $\operatorname{Spec}R$ is called specialization-closed if every prime ideal of $R$ containing some prime ideal in $X$ belongs to $X$. The following is an immediate consequence of Theorem 3.7.

Corollary 3.8**.**

Let $(R,\mathfrak{m})$ be a formally equidimensional local ring which is the homomorphic image of a Cohen-Macaulay ring. Let $X$ be a specialization-closed subset of $\operatorname{Spec}R$ such that $R_{\mathfrak{p}}$ is Gorenstein for all $\mathfrak{p}\in\operatorname{Spec}R\setminus X$. Assume $M$ is a horizontally linked $R$–module. The following statements are equivalent:

1. (i)

$\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(M))\subseteq X$* for $0<i<\dim R$.* 2. (ii)

$\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(\lambda M))\subseteq X$* for $0<i<\dim R$.*

In the following, we extend Theorem 3.3 to include a change of ring. First we need to recall some definitions and results from [13].

Definition 3.9**.**

Let $C$ be a semidualizing $R$–module. An $R$–module $M$ is called $\operatorname{\mathsf{G}_{\it C}}$-perfect if $\operatorname{\mathsf{grade}}_{R}(M)=\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(M)$. An $R$–module $M$ is called $\operatorname{\mathsf{G}_{\it C}}$-Gorenstein if $M$ is $\operatorname{\mathsf{G}_{\it C}}$-perfect and $\operatorname{\mathsf{Ext}}^{n}_{R}(M,C)$ is cyclic, where $n=\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R}(M)$. An ideal $I$ is called $\operatorname{\mathsf{G}_{\it C}}$-perfect (resp. $\operatorname{\mathsf{G}_{\it C}}$-Gorenstein) if $R/I$ is $\operatorname{\mathsf{G}_{\it C}}$-perfect (resp. $\operatorname{\mathsf{G}_{\it C}}$-Gorenstein) as $R$–module.

Note that if $I$ is a $\operatorname{\mathsf{G}_{\it C}}$-Gorenstein ideal of $\operatorname{\mathsf{G}_{\it C}}$-dimension $n$, then $\operatorname{\mathsf{Ext}}^{n}_{R}(R/I,C)\cong R/I$.

Theorem 3.10**.**

[13, Proposition 5]**. Let $K$ be a semidualizing $R$–module and let $I$ be a $\operatorname{\mathsf{G}_{\it K}}$-perfect ideal. Set $C=\operatorname{\mathsf{Ext}}^{\tiny{\operatorname{\mathsf{grade}}(I)}}_{R}(R/I,K)$. Then the following statements hold.

• (i)

$C$* is a semidualizing $R/I$–module.*

• (ii)

If $M$ is a $R/I$–module, then $\operatorname{\mathsf{G}_{\it K}\mathsf{-dim}}_{R}(M)<\infty$ if and only if $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R/I}(M)<\infty$, and if these dimensions are finite, then $\operatorname{\mathsf{G}_{\it K}\mathsf{-dim}}_{R}(M)=\operatorname{\mathsf{grade}}(I)+\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R/I}(M)$.

Corollary 3.11**.**

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring and let $C$ be a semidualizing $R$–module. Assume that $M$ is an $R$–module of finite $\operatorname{\mathsf{G}_{\it C}}$-dimension which is linked by a $\operatorname{\mathsf{G}_{\it C}}$-Gorenstein ideal $\mathfrak{c}$. Set $\overline{R}=R/\mathfrak{c}$. Suppose that $X$ is a subset of $\operatorname{Spec}\overline{R}$ which is stable under generalization. The following are equivalent:

1. (i)

$X\subseteq\widetilde{S}_{n}(M)$. 2. (ii)

$\operatorname{Att}_{\overline{R}}(\operatorname{H}^{i}_{\mathfrak{m}}(\lambda_{\overline{R}}M))\subseteq\operatorname{Spec}\overline{R}\setminus X$* for $\dim\overline{R}-n<i<\dim\overline{R}$.*

Proof.

As $R$ is Cohen-Macaulay and $\mathfrak{c}$ is $\operatorname{\mathsf{G}_{\it C}}$-perfect, $R/\mathfrak{c}$ is Cohen-Macaulay. By Theorem 3.10(ii), $\operatorname{\mathsf{G-dim}}_{R/\mathfrak{c}}(M)<\infty$. Now the assertion is clear by Theorem 3.3. ∎

For an $R$–module $M$, set $\operatorname{c}(M)=\sup\{i<\dim_{R}(M)\mid\operatorname{H}^{i}_{\mathfrak{m}}(M)\neq 0\}.$ In the following, we describe the attached prime ideals of $\operatorname{H}^{\tiny{\operatorname{c}(M)}}_{\mathfrak{m}}(M)$ for a non-Cohen-Macaulay horizontally linked $R$–module $M$. The following result should be compared with [9, Corollary 4.5].

Theorem 3.12**.**

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$. Assume that $M$ is an $R$–module of finite and positive Gorenstein dimension which is horizontally linked. Set $\operatorname{c}=\operatorname{c}(\lambda M)$. Then

[TABLE]

Proof.

As $\operatorname{H}_{\mathfrak{m}}^{\operatorname{c}}(\lambda M)$ is an Artinian $R$-module, it has a natural structure as an $\widehat{R}$-module. More precisely, it is an Artinian $\widehat{R}$-module. By Theorem 2.7(iv),

[TABLE]

Note that by local duality theorem and Lemma 3.1, $\widehat{M}$ is $(d-c)$-torsionfree $\widehat{R}$-module. It follows from Theorem 2.7(iii), local duality theorem and Lemma 3.2 that

[TABLE]

On the other hand, by [22, Proposition 9.A],

[TABLE]

It follows from (3.12.1), (3.12.2), (3.12.3) that

[TABLE]

Set $n=d-c$. As $M$ is $n$-torsionfree, by [3, Proposition 3.6], there exists the following exact sequence:

[TABLE]

where $P$ is projective $R$–module. Set $X=\Omega^{n+1}\mathsf{Tr}\Omega^{n+1}\mathsf{Tr}M\oplus P$ and $Y=\Omega^{n-1}\operatorname{\mathsf{Ext}}^{n+1}_{R}(\mathsf{Tr}M,R)$. Let $\mathfrak{p}\in\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{n}_{R}(\lambda M,R))$. Hence $\mathfrak{p}\in\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{n+1}_{R}(\mathsf{Tr}M,R))$ and so $\mathfrak{p}\notin S_{n+1}(M)$ by Theorem 2.4. Note that $\operatorname{\mathsf{depth}}R_{\mathfrak{p}}>n$. Therefore $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(Y_{\mathfrak{p}})=n-1$ and $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(X_{\mathfrak{p}})>n$. By localizing the exact sequence (3.12.5) at $\mathfrak{p}$, we conclude that $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})=n$.

Conversely, assume that $\mathfrak{p}\in S_{n}(M)\setminus S_{n+1}(M)$ and that $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})=n$. Hence by (2.3)(ii), $\operatorname{\mathsf{depth}}R_{\mathfrak{p}}>n$. By localizing the exact sequence (3.12.5) at $\mathfrak{p}$, we conclude that $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(Y_{\mathfrak{p}})=n-1$ and so $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(\operatorname{\mathsf{Ext}}^{n+1}_{R}(\mathsf{Tr}M,R)_{\mathfrak{p}})=0$. In other words, $\mathfrak{p}\in\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{n+1}_{R}(\mathsf{Tr}M,R))$. Now the assertion follows from (3.12.4). ∎

For a subset $X$ of $\operatorname{Spec}R$, we denote by $\overline{X}$ the closure of $X$ in the Zariski topology.

Corollary 3.13**.**

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$. Assume that $M$ is an $R$–module of finite and positive Gorenstein dimension which is horizontally linked. For a positive integer $n<d$, we have

[TABLE]

Proof.

First note that by [8, Theorem 4.2], for a prime ideal $\mathfrak{p}$ of $R$ we have

[TABLE]

Set $X=\left\{\mathfrak{p}\in S_{n}(M)\setminus S_{n+1}(M)\mid\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})=n\right\}$ and $Y=\overline{\underset{\tiny{d-n<i<d}}{\bigcup}\operatorname{Att}_{R}(\operatorname{H}^{i}_{\mathfrak{m}}(\lambda M))}$. Note that $\operatorname{Spec}R\setminus Y$ is stable under generalization. Hence, by Theorem 3.3, $\operatorname{Spec}R\setminus Y\subseteq S_{n}(M)$. Now let $\mathfrak{p}\in\operatorname{Att}_{R}(\operatorname{H}_{\mathfrak{m}}^{d-n}(\lambda M))\setminus Y$. Therefore $\mathfrak{p}\in S_{n}(M)$. Also by Theorem 2.8, $\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Att}_{R_{\mathfrak{p}}}(\operatorname{H}_{\mathfrak{p}R_{\mathfrak{p}}}^{\dim R_{\mathfrak{p}}-n}((\lambda M)_{\mathfrak{p}}))$. Hence, by (3.13.1), $\operatorname{c}_{R_{\mathfrak{p}}}(\lambda_{R_{\mathfrak{p}}}M_{\mathfrak{p}})=\dim R_{\mathfrak{p}}-n.$ It follows from Corollary 3.12 that $\mathfrak{p}R_{\mathfrak{p}}\notin S_{n+1}(M_{\mathfrak{p}})$ and $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})=n$. In other words, $\mathfrak{p}\in X$.

Conversely, if $\mathfrak{p}\in X$, then $\operatorname{c}_{R_{\mathfrak{p}}}(\lambda_{R_{\mathfrak{p}}}M_{\mathfrak{p}})=\dim R_{\mathfrak{p}}-n$ by (3.13.1). It follows from Corollary 3.12 that $\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Att}_{R_{\mathfrak{p}}}(\operatorname{H}_{\mathfrak{p}R_{\mathfrak{p}}}^{\dim R_{\mathfrak{p}}-n}((\lambda M)_{\mathfrak{p}}))$ and so $\mathfrak{p}\in\operatorname{Att}_{R}(\operatorname{H}_{\mathfrak{m}}^{d-n}(\lambda M))$ by Theorem 3.3. Assume contrarily that $\mathfrak{p}\in Y$. Hence there exists a prime ideal $\mathfrak{q}\subseteq\mathfrak{p}$ such that $\mathfrak{q}\in\operatorname{Att}_{R}(\operatorname{H}^{j}_{\mathfrak{m}}(\lambda M))$ for some integer $j$ with $d-n<j<d$. By Theorem 3.3, $\mathfrak{q}R_{\mathfrak{q}}\in\operatorname{Att}_{R_{\mathfrak{q}}}(\operatorname{H}_{\mathfrak{q}R_{\mathfrak{q}}}^{j-\dim R/\mathfrak{q}}((\lambda M)_{\mathfrak{q}}))$. As $j<d$, we have $j-\dim R/\mathfrak{q}<\dim R_{\mathfrak{q}}$. Note that $\mathfrak{q}\in S_{n}(M)$ and so by (3.13.1) $j-\dim R/\mathfrak{q}\leq\dim R_{\mathfrak{q}}-n$. Hence $j\leq d-n$ which is a contradiction. ∎

4. Linkage and Cohen-Macaulay modules

The first main theorem in the theory of linkage was due to C. Peskine and L. Szpiro. They proved that the Cohen-Macaulayness property is preserved under linkage over Gorenstein rings. Note that the above statement is no longer true if the base ring is Cohen-Macaulay but not Gorenstein. On the other hand, there is an interesting generalization of the Peskine-Szpiro Theorem, due to Martin [18]. Let $R$ be a Gorenstein local ring and let $I$ be Cohen-Macaulay ideal linked to $J$ by an ideal $\mathfrak{a}$, where $\mathfrak{a}$ is Cohen-Macaulay and generically Gorenstein. Then $J$ is Cohen-Macaulay if and only if $I+\mathfrak{b}$ is a Cohen-Macaulay ideal with the same height as $\mathfrak{b}$, where $\mathfrak{b}$ is an ideal of $R$, containing $\mathfrak{a}$, with $\mathfrak{b}/\mathfrak{a}$ equal to a canonical module of $\omega_{R/\mathfrak{a}}$ [18, Corolary 2.11].

In the theory of linkage of modules, Martsinkovsky and Strooker generalized the Peskine-Szpiro Theorem for stable modules over Gorenstein local rings [20, Proposition 8]. In this section, we study the theory of linkage for Cohen-Macaulay modules. As a consequence, we generalize the Martin’s result to the module case.

A semidualizing ideal is an ideal that is semidualizing as an $R$-module. Let $C$ be a semidualizing $R$–module. If $R$ is generically Gorenstein, then $C$ can be identified with an ideal of $R$ [34, Proposition 3.1]. Let $R$ be a Cohen-Macaulay local ring and let $\mathfrak{c}$ be a semidualizing ideal. Then $\mathfrak{c}$ is an ideal of height one or equals $R$ [34, Proposition 3.2].

Theorem 4.1**.**

Let $R$ be a generically Gorenstein and let $\mathfrak{c}$ be a semidualizing ideal of $R$. Assume that $M$ is a horizontally linked $R$–module. Then the following statements hold:

1. (i)

$\operatorname{\mathsf{G}_{\it\mathfrak{c}}\mathsf{-dim}}_{R}(M)=0$* if and only if $\operatorname{\mathsf{G}_{\it\mathfrak{c}}\mathsf{-dim}}_{R}(\mathfrak{c}\lambda M)=0$.* 2. (ii)

Assume that $M$ has finite $\operatorname{\mathsf{G}_{\it\mathfrak{c}}}$-dimension on $X^{n-1}(R)$. Then $M$ satisfies $\widetilde{S}_{n}$ if and only if $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathfrak{c}\lambda M,\mathfrak{c})=0$ for $0<i<n$. 3. (iii)

Assume that $R$ is Cohen-Macaulay and that $\operatorname{\mathsf{G}_{\it\mathfrak{c}}\mathsf{-dim}}_{R}(M)<\infty$. Then $M$ is Cohen-Macaulay if and only if $\mathfrak{c}\lambda M$ is so.

Proof.

Let $\cdots\rightarrow F_{1}\rightarrow F_{0}\overset{f}{\rightarrow}M\rightarrow 0$ be the minimal free resolution of $M$. From the exact sequence $(F_{0})^{*}\overset{g}{\rightarrow}\lambda M\rightarrow 0$ we get the following exact sequence $(F_{0})^{*}\otimes_{R}\mathfrak{c}\overset{g\otimes\mathfrak{c}}{\rightarrow}\lambda M\otimes_{R}\mathfrak{c}\rightarrow 0$. Let $\phi=h\circ(g\otimes\mathfrak{c})$, where $h$ is the natural epimorphism $\lambda M\otimes_{R}\mathfrak{c}\twoheadrightarrow\mathfrak{c}\lambda M$. We denote by $\psi$ the tensor evaluation isomorphism $(F_{0})^{*}\otimes_{R}\mathfrak{c}\overset{\cong}{\rightarrow}\operatorname{\mathsf{Hom}}_{R}(F_{0},\mathfrak{c})$. Dualize the free resolution of $M$, into $\mathfrak{c}$ to obtain the exact sequence

[TABLE]

Set $\lambda_{\mathfrak{c}}(M):=\operatorname{Coker}(\operatorname{\mathsf{Hom}}(f,\mathfrak{c}))$. We denote by $\alpha$ the epimorphism $\operatorname{\mathsf{Hom}}_{R}(F_{0},\mathfrak{c})\twoheadrightarrow\lambda_{\mathfrak{c}}(M)$. Now it is easy to check that $\phi\circ\psi^{-1}\circ\operatorname{\mathsf{Hom}}(f,\mathfrak{c})=0=\alpha\circ\psi\circ\iota$, where $\iota:\ker\phi\hookrightarrow(F_{0})^{*}\otimes_{R}\mathfrak{c}$ is the natural inclusion. Therefore we obtain the following commutative diagram

[TABLE]

with exact rows. Hence we obtain the following isomorphism:

[TABLE]

(i). From the exact sequence (4.1.1) we get the following short exact sequence:

[TABLE]

As $M$ is a syzygy module, it is easy to see that $\operatorname{\mathsf{Ext}}^{1}_{R}(\mathsf{Tr}_{\mathfrak{c}}(M),\mathfrak{c})=0$. Now the assertion follows from 2.3(iv), (4.1.2) and the exact sequence (4.1.3).

(ii). By [9, Proposition 2.4], $M$ satisfies $\widetilde{S}_{n}$ if and only if $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}_{\mathfrak{c}}(M),\mathfrak{c})=0$ for $1\leq i\leq n$. As $M$ is horizontally linked, it satisfies $\widetilde{S}_{1}$ and so $\operatorname{\mathsf{Ext}}^{1}_{R}(\mathsf{Tr}_{\mathfrak{c}}(M),\mathfrak{c})=0$. The exact sequence (4.1.3) induces the following isomorphism $\operatorname{\mathsf{Ext}}^{i}_{R}(\lambda_{\mathfrak{c}}(M),\mathfrak{c})\cong\operatorname{\mathsf{Ext}}^{i+1}_{R}(\mathsf{Tr}_{\mathfrak{c}}M,\mathfrak{c})$ for all $i>0$. Now the assertion follows from (4.1.2).

(iii). If $M$ is Cohen-Macaulay, then $\mathfrak{c}\lambda M$ is Cohen-Macaulay by 2.3(ii) and part (i).

Conversely, assume $\mathfrak{c}\lambda M$ is Cohen-Macaulay. Suppose contrarily that $M$ is not Cohen-Macaulay. Hence $\operatorname{\mathsf{G}_{\it\mathfrak{c}}\mathsf{-dim}}_{R}(M)>0$. Set $n=\min\{i>0\mid\operatorname{\mathsf{Ext}}^{i}_{R}(M,\mathfrak{c})\neq 0\}$. As $\operatorname{\mathsf{Ext}}^{i}_{R}(M,\mathfrak{c})=0$ for all $0<i<n$, applying functor $(-)^{\triangledown}=\operatorname{\mathsf{Hom}}_{R}(-,\mathfrak{c})$ to the minimal free resolution of $M$ implies the following exact sequences:

[TABLE]

[TABLE]

Now let $\mathfrak{p}\in\operatorname{Ass}_{R}(\operatorname{\mathsf{Ext}}^{n}_{R}(M,\mathfrak{c}))$. It follows from the exact sequence (4.1.5) that $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}((\mathsf{Tr}_{\mathfrak{c}}\Omega^{n-1}(M)_{\mathfrak{p}})=0$. As $\operatorname{\mathsf{Ext}}^{n}_{R}(M,\mathfrak{c})_{\mathfrak{p}}\neq 0$, by 2.3(ii), (iii), $\operatorname{\mathsf{depth}}R_{\mathfrak{p}}-\operatorname{\mathsf{depth}}(M_{\mathfrak{p}})=\operatorname{\mathsf{G}_{\it{\mathfrak{c}}_{\mathfrak{p}}}\mathsf{-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq n$. Since $M$ is a first syzygy, we conclude that $\operatorname{\mathsf{depth}}R_{\mathfrak{p}}>n$. It follows from the exact sequence (4.1.4) that $\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(\lambda_{\mathfrak{c}}(M)_{\mathfrak{p}})=n$ which is a contradiction, because $\lambda_{\mathfrak{c}}(M)\cong\mathfrak{c}\lambda M$ is a maximal Cohen-Mcaulay module by our assumption. ∎

In the following, we extend Theorem 4.1 to include a change of ring.

Corollary 4.2**.**

Let $R$ be a Cohen-Macaulay local ring and let $K$ be a semidualizing $R$–module. Assume $\mathfrak{a}$ is a $\operatorname{\mathsf{G}_{\it K}}$-perfect ideal of $R$ which is generically Gorenstein. Set $C=\operatorname{\mathsf{Ext}}^{m}_{R}(R/\mathfrak{a},K)$, where $m=\operatorname{\mathsf{grade}}(\mathfrak{a})$ and write $C\cong\mathfrak{c}/\mathfrak{a}$. Assume that $M$ and $N$ are $R$–modules such that $M\underset{\mathfrak{a}}{\sim}N$. Then the following statements hold:

1. (i)

$M$* is $\operatorname{\mathsf{G}_{\it K}}$-perfect if and only if $\mathfrak{c}N$ is $\operatorname{\mathsf{G}_{\it K}}$-perfect.* 2. (ii)

Assume that $M$ has finite $\operatorname{\mathsf{G}_{\it K}}$-dimension. Then $M$ satisfies $(S_{n})$ if and only if $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathfrak{c}N,K)=0$ for $m<i<n+m$. Also, $M$ is Cohen-Macaulay if and only if $\mathfrak{c}N$ is so.

Proof.

(i). By Theorem 3.10(ii) and [31, Lemma 3.16], $M$ is $\operatorname{\mathsf{G}_{\it K}}$-perfect if and only if $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R/\mathfrak{a}}(M)=0$. Note that $\operatorname{\mathsf{grade}}(\mathfrak{c}N)=\operatorname{\mathsf{grade}}(N)$. Again by using Theorem 3.10(ii) and [31, Lemma 3.16], we have $\mathfrak{c}N$ is $\operatorname{\mathsf{G}_{\it K}}$-perfect if and only if $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R/\mathfrak{a}}(\mathfrak{c}N)=0$. Now the assertion follows from Theorem 4.1(i).

(ii). By Theorem 3.10(ii), $\operatorname{\mathsf{G}_{\it C}\mathsf{-dim}}_{R/\mathfrak{a}}(M)<\infty$. As $\mathfrak{a}$ is $\operatorname{\mathsf{G}_{\it K}}$-perfect, $R/\mathfrak{a}$ is a Cohen-Macaulay ring. By [13, Corollary], $\operatorname{\mathsf{Ext}}^{i}_{R/\mathfrak{a}}(\mathfrak{c}N,C)\cong\operatorname{\mathsf{Ext}}^{i+m}_{R}(\mathfrak{c}N,K)$ for all $i\geq 0$. Now the assertion follows from Theorem 4.1(ii), (iii). ∎

The following is a generalization of [18, Corollary 2.11].

Corollary 4.3**.**

Let $R$ be a Cohen-Macaulay local ring and let $K$ be a semidualizing $R$–module. Assume $\mathfrak{a}$ is a $\operatorname{\mathsf{G}_{\it K}}$-perfect ideal of $R$ which is generically Gorenstein. Set $C=\operatorname{\mathsf{Ext}}^{n}_{R}(R/\mathfrak{a},K)$, where $n=\operatorname{\mathsf{grade}}(\mathfrak{a})$ and write $C\cong\mathfrak{c}/\mathfrak{a}$. Assume that $M$ is an $R$–module of finite $\operatorname{\mathsf{G}_{\it K}}$-dimension and that $N$ is a Cohen-Macaulay $R$–module which is linked to $M$ by the ideal $\mathfrak{a}$. Then $M$ is Cohen-Macaulay if and only if $\operatorname{\mathsf{depth}}_{R}(N/\mathfrak{c}N)\geq\dim(R/\mathfrak{a})-1$.

Proof.

Consider the following exact sequence

[TABLE]

Now the assertion follows from Theorem 4.1(ii) and the exact sequence (4.3.1). ∎

It should be noted that Theorem B is a special case of Corollary 4.3.

Corollary 4.4**.**

Let $R$ be a Cohen-Macaulay local ring with canonical module $\omega_{R}$ and let $\mathfrak{a}$ be a Cohen-Macaulay ideal of $R$ which is generically Gorenstein. Write $\omega_{R/\mathfrak{a}}\cong\mathfrak{c}/\mathfrak{a}$. Assume that $I$ and $J$ are ideals of $R$ linked by the ideal $\mathfrak{a}$. The following statements hold:

1. (i)

$R/I$* satisfies $(S_{n})$ if and only if $\operatorname{H}^{i}_{\mathfrak{m}}(\mathfrak{c}/\mathfrak{c}\cap J)=0$ for $\dim(R/\mathfrak{a})-n<i<\dim(R/\mathfrak{a})$. In particular, $R/I$ is Cohen-Macaulay if and only if $\mathfrak{c}/\mathfrak{c}\cap J$ is so.* 2. (ii)

Assume $R/I$ is Cohen-Macaulay. Then $R/J$ is Cohen-Macaulay if and only if $\operatorname{\mathsf{depth}}_{R}(R/(I+\mathfrak{c}))\geq\dim R/\mathfrak{a}-1$.

Proof.

This follows immediately from Corollaries 4.2, 4.3 and the local daulity theorem. ∎

Let $n$ be a positive integer. An $R$–module $M$ is said to be an $n$th syzygy if there exists an exact sequence

[TABLE]

where $P_{0},\cdots,P_{n-1}$ are projective. By convention, every module is a [math]th syzygy.

Theorem 4.5**.**

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ with canonical module $\omega_{R}$. Assume that $M$ is a horizontally linked $R$–module and that $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$ for all $\mathfrak{p}\in\operatorname{Spec}R-\{\mathfrak{m}\}$. Then the following conditions are equivalent:

1. (i)

$M\otimes_{R}\omega_{R}$* is maximal Cohen-Macaulay.* 2. (ii)

$\lambda M$* is *$(d+1)$th syzygy module.

Proof.

By [30, Theorem 10.62], there is a third quadrant spectral sequence:

[TABLE]

As $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$ for all $\mathfrak{p}\in\operatorname{Spec}R-\{\mathfrak{m}\}$, $\operatorname{\mathsf{Tor}}_{i}^{R}(M,\omega_{R})$ has finite length for all $i>0$ (see 2.5). Therefore, $\operatorname{E}^{p,q}_{2}=0$ when $q>0$ and $p<d$ and so we obtain the following isomorphisms:

[TABLE]

(i)$\Rightarrow$(ii). It follows from (4.5.1) and our assumption that $\operatorname{\mathsf{Ext}}^{i}_{R}(M,R)=0$ for all $1\leq i\leq d$. As $M$ is horizontally linked, $M\cong\lambda^{2}M\cong\Omega\mathsf{Tr}\lambda M$. Therefore, $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}\lambda M,R)=0$ for all $1\leq i\leq d+1$ and so $\lambda M$ is $(d+1)$th syzygy module by [21, Proposition 11].

(ii)$\Rightarrow$(i). By [8, Proposition 2.6] and our assumption, $\operatorname{\mathsf{Ext}}^{i}_{R}(M,R)_{\mathfrak{p}}=0$ for all $\mathfrak{p}\in\operatorname{Spec}R-\{\mathfrak{m}\}$ and all $1\leq i\leq\operatorname{\mathsf{depth}}R_{\mathfrak{p}}$. Hence, by (2.3)(iii), $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})=0$ and so $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}((\lambda M)_{\mathfrak{p}})=0$ for all $\mathfrak{p}\in\operatorname{Spec}R-\{\mathfrak{m}\}$. It follows from [21, Theorem 43] that $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}\lambda M,R)=0$ for all $1\leq i\leq d+1$. As $M$ is horizontally linked, $M\cong\Omega\mathsf{Tr}\lambda M$. Therefore, $\operatorname{\mathsf{Ext}}^{i}_{R}(M,R)=0$ for all $1\leq i\leq d$. Now the assertion is clear by (4.5.1). ∎

Here is a characterization of Gorenstein rings in terms of linkage.

Proposition 4.6**.**

Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay local ring of dimension $d>1$. Then the following statements are equivalent:

1. (i)

$R$* is Gorenstein.* 2. (ii)

The Cohen-Macaulayness property is presereved under horizontal linkage.

Proof.

(i)$\Rightarrow$(ii). Follows from [20, Proposition 8]. (ii)$\Rightarrow$(i). By [4, Corollary 1.2.5], $\Omega^{d+1}k$ is stable and so it is horizontally linked (see 2.2). It follows from the assumption that $\lambda(\Omega^{d+1}k)$ is maximal Cohen-Macaulay. Hence $\operatorname{\mathsf{depth}}_{R}(\mathsf{Tr}\Omega^{d+1}k)\geq d-1\geq 1$. Now consider the following exact sequence:

[TABLE]

As $\operatorname{\mathsf{Ext}}^{d+2}_{R}(k,R)$ has finite length and $\operatorname{\mathsf{depth}}_{R}(\mathsf{Tr}\Omega^{d+1}k)>0$, we conclude that $\operatorname{\mathsf{Ext}}^{d+2}_{R}(k,R)=0$. In other words, $R$ is Gorenstein. ∎

5. Duality between local cohomology modules of linked module

Recall that an $R$–module $M$ of dimension $d\geq 1$ is called a

• generalized Cohen-Macaulay* module if $\ell(\operatorname{H}^{i}_{\mathfrak{m}}(M))<\infty$ for all $i$, $0\leq i\leq d-1$, where $\ell$ denotes the length. Let $R$ be a Gorenstein local ring and let $\mathfrak{a}$, $\mathfrak{b}$ be ideals of $R$ linked by a Gorenstein ideal $\mathfrak{c}$. Assume that $R/\mathfrak{a}$ is generalized Cohen-Macaulay. Schenzel established an interesting duality between local cohomology modules of $R/\mathfrak{a}$ and $R/\mathfrak{b}$. More precisely, he proved that

[TABLE]

for $0<i<d$, where $d=\dim R/\mathfrak{a}=\dim R/\mathfrak{b}$ [32, Corollary 3.3]. Martsinkovsky and Strooker generalized the above result for all generalized Cohen-Macaulay module which is linked by a Gorenstein ideal [20, Theorem 11]. In this Section, we generalize the above result to modules which satisfy Serre’s condition $(S_{n})$ on the punctured spectrum of $R$.

For an ideal $\mathfrak{a}$ of $R$, we denote by $V(\mathfrak{a})$ the set of all prime ideals of $R$ containing $\mathfrak{a}$. Let $X$ be a closed subset of $\operatorname{Spec}R$. Then $X=V(\mathfrak{a})$ for some ideal $\mathfrak{a}$ of $R$. We call such an ideal $\mathfrak{a}$ the defining ideal of $X$. This is uniquely determined up to radical.

Theorem 5.1**.**

Let $U$ be an open subset of $\operatorname{Spec}R$ and let $M$ be an $R$–module such that $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$ for all $\mathfrak{p}\in U$. Assume that $U\subseteq\widetilde{S}_{n}(M)$ for some $0<n\leq\operatorname{\mathsf{grade}}(\mathfrak{a})$, where $\mathfrak{a}$ is the defining ideal of the complement of $U$ in $\operatorname{Spec}R$. Then $\operatorname{H}^{i}_{\mathfrak{a}}(M)\cong\operatorname{\mathsf{Ext}}^{i+1}_{R}(\mathsf{Tr}M,R)$ for $0\leq i<n.$ In particular, $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is finitely generated for $0\leq i<n$.

Proof.

As $M_{\mathfrak{p}}$ satisfies $\widetilde{S}_{n}$, $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}M,R)_{\mathfrak{p}}=0$ for all $1\leq i\leq n$ and all $\mathfrak{p}\in\operatorname{Spec}(R)\setminus\operatorname{V}(\mathfrak{a})$ by Theorem 2.4. Therefore, $\underset{1\leq i\leq n}{\bigcup}\operatorname{Supp}_{R}(\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}M,R))\subseteq\operatorname{V}(\mathfrak{a})$ and so $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}M,R)$ is $\mathfrak{a}$-torsion for $1\leq i\leq n$. Consider the following exact sequence:

[TABLE]

where $X_{i}\approx\Omega\mathsf{Tr}\Omega^{i}\mathsf{Tr}M$. Since $\operatorname{\mathsf{Ext}}^{i}_{R}(\mathsf{Tr}M,R)$ is $\mathfrak{a}$-torsion for $1\leq i\leq n$, applying the functor $\Gamma_{\mathfrak{a}}(-)$ to the exact sequence (5.1.1), we get

[TABLE]

and also

[TABLE]

because $\operatorname{\mathsf{grade}}(\mathfrak{a})>0$. On the other hand, if $n>1$, then

[TABLE]

Now by using (5.1.2), (5.1.3) and (5.1.4) we obtain the result. ∎

The following is an immediate consequence of Theorem 5.1.

Corollary 5.2**.**

Let $U$ be an open subset of $\operatorname{Spec}R$ and let $M$ be a horizontally linked $R$–module such that $\operatorname{\mathsf{G-dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$ for all $\mathfrak{p}\in U$. Assume that $U\subseteq\widetilde{S}_{n}(M)$ for some $1<n\leq\operatorname{\mathsf{grade}}(\mathfrak{a})$, where $\mathfrak{a}$ is the defining ideal of the complement of $U$ in $\operatorname{Spec}R$. Then $\operatorname{H}^{i}_{\mathfrak{a}}(M)\cong\operatorname{\mathsf{Ext}}^{i}_{R}(\lambda M,R)\text{ for }0<i<n.$

Let $R$ be a Cohen-Macaulay local ring. An $R$–module $M$ is generalized Cohen-Macaulay if and only if $M_{\mathfrak{p}}$ is a Cohen-Macaulay $R_{\mathfrak{p}}$–module for all $\mathfrak{p}\in\operatorname{Spec}(R)-\{\mathfrak{m}\}$ (see [33, Lemma 1.2 , Lemma 1.4]). Therefore the following result can be viewed as a generalization of [20, Theorem 10].

Corollary 5.3**.**

Let $(R,\mathfrak{m},k)$ be a Gorenstein local ring of dimension $d>1$ and let $M$ be a horizontally linked $R$–module. Assume $n$ is a positive integer and that $M_{\mathfrak{p}}$ satisfies $(S_{n})$ for all $\mathfrak{p}\in\operatorname{Spec}R\setminus\{\mathfrak{m}\}$. Then $\operatorname{H}^{i}_{\mathfrak{m}}(M)\cong\operatorname{\mathsf{Hom}}_{R}(\operatorname{H}^{d-i}_{\mathfrak{m}}(\lambda M),\operatorname{E}_{R}(k))\text{ for }0<i<n$, where $\operatorname{E}_{R}(k)$ is the injective envelope of $k$.

Proof.

Follows from Corollary 5.2 and the local duality theorem. ∎

We end this section by the following result which extend Corollary 5.3 to include a change of ring.

Corollary 5.4**.**

Let $(R,\mathfrak{m},k)$ be a Gorenstein local ring, $\mathfrak{c}$ a Gorenstein ideal of $R$, $\overline{R}=R/\mathfrak{c}$, and $d=\dim\overline{R}>1$. Assume that $M$ is an $R$–module which is linked by the ideal $\mathfrak{c}$ and that $M_{\mathfrak{p}}$ satisfies $(S_{n})$ for all $\mathfrak{p}\in\operatorname{Spec}\overline{R}\setminus\{\overline{\mathfrak{m}}\}$. Then $\operatorname{H}^{i}_{\mathfrak{m}}(M)\cong\operatorname{\mathsf{Hom}}_{R}(\operatorname{H}^{d-i}_{\mathfrak{m}}(\lambda_{\overline{R}}M),\operatorname{E}_{R}(k))\text{ for }0<i<n$.