$I$-Cohen Macaulay modules

TL;DR

This paper introduces and studies the properties of $I$-Cohen Macaulay modules, a generalization of Cohen Macaulay modules, revealing their structure and providing various characterizations.

Contribution

It defines $I$-Cohen Macaulay modules and explores their properties, establishing analogies with Cohen Macaulay modules and offering new characterizations.

Findings

$I$-Cohen Macaulay modules share properties with Cohen Macaulay modules.

The paper provides multiple characterizations of $I$-Cohen Macaulay modules.

Structural insights into $I$-Cohen Macaulay modules are presented.

Abstract

A finitely generated module $M$ over a commutative Noetherian ring $R$ is called an $I$-Cohen Macaulay module, if \[ \grade(I,M) + \dim(M/IM)= \dim(M), \] where $I$ is a proper ideal of $R$. The aim of this paper is to study the structure of this class of modules. It is discovered that $I$-Cohen Macaulay modules enjoy many interesting properties which are analogous to those of Cohen Macaulay modules. Also, various characterizations of $I$-Cohen Macaulay modules are presented here.

Equations

$$\mathrm{grade}(I,M)+\dim (M/IM)=\dim (M),$$

$$\mathrm{grade}(\mathfrak{p},M)=\mathrm{grade}(\mathfrak{p}{R}_{\mathfrak{p}},M_{\mathfrak{p}}),$$

$$r+\dim (M/(x_1,x_2,...,x_r)M)\ge \dim (M).$$

$$\mathrm{grade}(I,M)+\dim (M/IM)\le r+\dim (M/(x_1,x_2,...,x_r)M)=\dim (M).$$

$$\mathrm{grade}(I,R)+\dim (R/I)\le 0pt(I)+\dim (R/I)\le \dim R.$$

$$\mathrm{grade}(I,M)+{\dim }_R(M/IM)={\dim }_R(M)$$

$$R/I=\frac{A[y][x]}{<Ux-1>}\cong A_U[y],$$

$$\mathrm{grade}(I,R)+\dim (R/I)=0pt(I)+\dim R/I<\dim R.$$

$$\mathrm{grade}(I,R)+\dim (R/I)=\dim R.$$

$$\dim (M/IM)=\dim (M/(x_1,x_2,\cdots x_r)M).$$

$${\varphi }_1:R\to R/{\mathrm{Ann}}_R(I)\text{ and }{\varphi }_2:M\to M/{\mathrm{Ann}}_R(I)M.$$

$$\overline{r}\cdot \overline{x}=\overline{0}\text{ and }x\notin {\mathrm{Ann}}_R(I)M.$$

$$r(Ix)=\sum _{i=1}^s(b_{i}I)y_i=0$$

$${\varphi }_2(M)/I{\varphi }_2(M)\cong M/IM$$

$$\mathrm{grade}({\varphi }_1(I),{\varphi }_2(M))$$

$$\ge \dim ({\varphi }_2(M))-\dim ({\varphi }_2(M)/I{\varphi }_2(M))$$

$$\ge \mathrm{grade}({\varphi }_1(I),{\varphi }_2(M)),\text{ see Equation (refe4)}.$$

$${\dim }_{R}M/IM={\dim }_S(M{\otimes }_{R}N)/I(M{\otimes }_{R}N).$$

$${\dim }_{R}M={\dim }_S(M{\otimes }_{R}N).$$

$$\frac{N/IN}{(\mathfrak{m}/I)(N/IN)}\cong N/\mathfrak{m}N.$$

$$\mathrm{grade}(I,M)={\dim }_{S}M{\otimes }_{R}N-{\dim }_{R}M/IM\text{ and }$$

$${\dim }_{S/IS}(M/IM){\otimes }_{R/I}(N/IN)={\dim }_{R/I}(M/IM)+{\dim }_{S/IS}\frac{N/IN}{(\mathfrak{m}/I)(N/IN)},$$

$$(M/IM){\otimes }_{R/I}(N/IN)\cong (M{\otimes }_{R}N)/I(M{\otimes }_{R}N).$$

$${\dim }_S(M{\otimes }_{R}N)/I(M{\otimes }_{R}N)={\dim }_R(M/IM)+{\dim }_S\frac{N/IN}{(\mathfrak{m}/I)(N/IN)}.$$

$$\mathrm{grade}(I,M)\ge {\dim }_S(M{\otimes }_{R}N)-{\dim }_S(M{\otimes }_{R}N)/I(M{\otimes }_{R}N)\ge \mathrm{grade}(IS,M{\otimes }_{R}N).$$

$${\dim }_{R_{\mathfrak{p}}}{M}_{\mathfrak{p}}/I{M}_{\mathfrak{p}}={\dim }_{S_{\mathfrak{q}}}\frac{(M{\otimes }_{R}N{)}_{\mathfrak{p}}}{I{S}_{\mathfrak{q}}(M{\otimes }_{R}N{)}_{\mathfrak{p}}}.$$

$${\dim }_{R_{\mathfrak{p}}}{M}_{\mathfrak{p}}={\dim }_{S_{\mathfrak{q}}}(M{\otimes }_{R}N{)}_{\mathfrak{p}}.$$

$$\mathrm{grade}(I,M)+\dim (\frac{M}{IM})\le \dim (M)=\mathrm{grade}(J,M)+\dim (\frac{M}{JM})\le \mathrm{grade}(I,M)+\dim (\frac{M}{IM})$$

$$\frac{\acute{M}}{\acute{I}\acute{M}}\cong \frac{M}{IM},$$

$$\mathrm{grade}(\frac{I}{<x_1,x_2,\cdots x_r>},\frac{IM}{<x_1,x_2,\cdots x_r>M})=\mathrm{grade}(I,M)-r,$$

$${\dim }_{R}M={\dim }_{R}R/\mathfrak{q},\text{ for some }\mathfrak{q}\in {\mathrm{Ass}}_R(M)\text{ such that }\mathfrak{q}\subseteq \mathfrak{p}.$$

Full text

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where $I$ is a proper ideal of $R$. The aim of this paper is to study the structure of this class of modules. It is discovered that $I$-Cohen Macaulay modules enjoy many interesting properties which are analogous to those of Cohen Macaulay modules. Also, various characterizations of $I$-Cohen Macaulay modules are presented here.

1. Introduction

The notions of Cohen Macaulay modules, as well as almost Cohen Macaulay modules are well established in commutative algebra. This paper is devoted to introduce and study the concept of $I$-Cohen Macaulay modules.

In the structure of $I$-Cohen Macaulay modules, we confine our attention entirely to arbitrary Noetherian ring which is not necessarily local. It is really interesting to establish the theory of $I$-Cohen Macaulay modules. As might be expected, much of the motivation for our work comes from rapidly developing theory of commutative algebra.

A finitely generated module $M$ over a Noetherian ring $R$ is called an almost Cohen Macaulay module, if

[TABLE]

for every $\mathfrak{p}\in\operatorname{Supp}(M)$. Basically a flaw (If $S$ is commutative Noetherian ring and $M$ any finitely generated $S$-module, then $\operatorname{grade}(\mathfrak{p},M)=\operatorname{grade}(\mathfrak{p}R_{\mathfrak{p}},M_{\mathfrak{p}})\hskip 5.69054pt$ for every $\mathfrak{p}\in\operatorname{Supp}(M)$ ) [11, 15.C, page 97] was corrected in [12, Exercise 16.5].

This led to the study of almost Cohen Macaulay rings and modules, first studied by Y. Han in 1998 (he referred it as $D$-ring) and M. Kang in 2001 [9]. Some fascinating examples are given in [10] by Kang. Later, this class of modules is studied by several authors (see [3], [7], [8], [13], [14] and [15].)

Ionescu discovered that how an almost Cohen-Macaulay module behaves when tensoring by a flat module. In this paper, this phenomena for $I$-Cohen Macaulay modules is discussed.

Many properties of $I$-Cohen Macaulay modules including the results about perseverance of $I$-Cohen Macaulayness in polynomial ring, formal power series ring and completion of the ring are given.

As for every proper ideal $I$ of $R$, a Cohen Macaulay module is $I$-Cohen Macaulay, it was mysterious (at least to the authors), that under what conditions an $I$-Cohen Macaulay module would be a Cohen Macaulay module? In this paper, a characterization regarding this issue is also given.

Throughout this paper, $R$ will be denoting a commutative Noetherian ring and $M$ a non-zero finitely generated $R$-module, unless otherwise specified. Besides above mentioned characterizations some other results of this note are following.

2. Main results

A description of results regarding dimension and grade of modules is being given. For detailed study, see [2], [11] and [12].

Let $R$ be a local ring and $I$ a proper ideal of $R$. Then,

[TABLE]

where $x_{1},x_{2},...,x_{r}\in I$. If $x_{1},x_{2},...,x_{r}\in I$ is a maximal $M$-regular sequence, then,

[TABLE]

Moreover, if $M=R$, then,

[TABLE]

Definition 2.1**.**

Let $R$ be a ring and $I$ a proper ideal of $R$. Then an $R$-module $M$ is said to be $I$-Cohen Macaulay, if the following equality holds:

[TABLE]

Remark 2.2*.*

If $I$ is a proper ideal of a local ring $R$. Then $M$ is $I$-Cohen Macaulay for each of the following cases:

• (1)

If $I$ is generated by an $M$-regular sequence, see Equation (2.1).

• (2)

$M$ is Cohen Macaulay, see [2, Theorem 2.1.2].

Remark 2.2 will not hold, provided that $R$ is not local. Moreover, it is shown that there is no relation between almost Cohen Macaulay module and $I$-Cohen Macaulay module, even if $I$ is a prime ideal.

Example 2.3**.**

$(1)$ Let $R=A[x,y]$ where, $A=k[|U,V|]_{<U>}$ and $k$ a field. Note that $A$ is a DVR and $R$ a non-local Cohen Macaulay ring with $\dim R=3$. Consider the ideal $I=<Ux-1>$ of $R$. Then $\operatorname{grade}(I,R)=0pt(I)=1$. There exists the following isomorphism

[TABLE]

Now $A$ is a DVR, so $A_{U}\cong k(|U,V|)$. Hence, $A_{U}[y]$ is a one dimensional P.I.D. Then $I$ is a prime ideal and $\dim R/I=\dim A_{U}[y]=1$. So,

[TABLE]

This proves that $R$ is not $I$-Cohen Macaulay.

$(2)$ Let $k$ be a field, $A=k[x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}]$ and $R=A/J$, where $J=<x_{1},x_{2},x_{3}>\cap<y_{1},y_{2},y_{3}>$. By [10, Example, p-4], $S$ is not an almost Cohen Macaulay ring. Let $I=<x_{1},x_{2},x_{3}>R$. Note that $\operatorname{Ass}_{R}(R)=\{I,<y_{1},y_{2},y_{3}>\}$. Hence, $\operatorname{grade}(I,R)=0$ and $\dim R/I=3.$ So,

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Remark 2.4*.*

$(1)$ If $R$ is $I$-Cohen Macaulay, then $\operatorname{grade}(I,R)=0pt(I)$, but converse is not true, see Example 2.3$(1)$,

$(2)$ If $R$ is local, $M$ an $I$-Cohen Macaulay module and $x_{1},x_{2},\cdots,x_{r}$ a maximal $M$-regular sequence in $I$. Form Equation (2.1), it follows that

[TABLE]

Let $I$ be a non-zero proper ideal in $R$ and $M$ an $R$-module. In the next result, it will be shown that regular sequences are persevered under the following natural onto ring and module homomorphisms resp.:

[TABLE]

Proposition 2.5**.**

With the previous notion, assume that $r_{1},r_{2},\dots,r_{n}\in I$ is an $M$-sequence such that $\phi_{1}({r_{i}})\notin\ker(\phi_{1})$, for all $i=1,\dots,n$ (e.g. ${r_{1}},{r_{2}},\dots,{r_{n}}$ is $R$-regular). Then $\phi_{1}({r_{1}}),\phi_{1}({r_{2}}),\dots,\phi_{1}({r_{n}})\in\phi_{1}(I)$ is an $\phi_{2}(M)$-sequence.

Proof.

Let $\overline{R}=\phi_{1}(R)$ and $\overline{M}=\phi_{2}(M)$. For $n=1$, suppose that $r\in I$ is $M$-regular. Let $\overline{r}=r+\operatorname{Ann}_{R}(I)$ and $\overline{x}=x+\operatorname{Ann}_{R}(I)M\in\overline{M}$ such that

[TABLE]

It implies that $rx=\sum_{i=1}^{s}b_{i}y_{i}$ with $b_{i}\in\operatorname{Ann}_{R}(I)$ and $y_{i}\in M$ for all $i=1,\dots s$. Then

[TABLE]

In particular, $r(rx)=0$. By regularity of $r$ on $M$, $x=0$ a contradiction. So, $\overline{r}\in\phi_{1}(I)$ is $\overline{M}$-regular. Hence, the result follows by induction. ∎

Corollary 2.6**.**

Assume that $I$ is a non-zero proper ideal in a local ring $R$ and $\operatorname{Ann}_{R}(I)\subseteq I$. For, any $R$-module $M$, the following conditions are equivalent:

• (1)

$M$* is an $I$-Cohen Macaulay module.*

• (2)

$\phi_{2}(M)$* is $I\phi_{2}(M)$-Cohen Macaulay as a $\phi_{1}(R)$-module, $\operatorname{grade}(I,M)=\operatorname{grade}(\phi_{1}(I),\phi_{2}(M))$ and $\dim M=\dim\phi_{2}(M)$.*

Proof.

Let $M$ be an $I$-Cohen Macaulay module. Since

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By Theorem 2.5, $\operatorname{grade}(I,M)\leq\operatorname{grade}(\phi_{1}(I),\phi_{2}(M))$. It implies that

[TABLE]

This proves all the statements in $(2)$. The converse is obvious. ∎

If $f:(R,\mathfrak{m})\to(S,\mathfrak{n})$ is a local ring map of Noetherian local rings. Suppose $N$ is an $R$-flat finitely generated $S$-module. In [2, Theorem 2.1.7] and [7, Proposition 2.2], the authors investigated behavior of Cohen macaulay and almost Cohen Macaulay modules upon tensoring $f$ with $N$. Next result discuses the behavior of $I$-Cohen Macaulay modules, when $f$ is tensoring with $N$.

Theorem 2.7**.**

With the above notion, assume that $I$ is any proper ideal of $R$. Let $M$ be an $R$-module and $M\otimes_{R}N\neq I(M\otimes_{R}N)$. Then following conditions are equivalent:

• (1)

$M$* is $I$-Cohen Macaulay and $\dim_{S}N/\mathfrak{m}N=0$.*

• (2)

$M\otimes_{R}N$* is $IS$-Cohen Macaulay, $\operatorname{grade}(I,M)=\operatorname{grade}(IS,M\otimes_{R}N)$ and*

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• (3)

$M\otimes_{R}N$* is $IS$-Cohen Macaulay, $\operatorname{grade}(I,M)=\operatorname{grade}(IS,M\otimes_{R}N)$ and*

[TABLE]

Proof.

Only the implications $(1)\Rightarrow(2)$ and $(1)\Rightarrow(3)$ are proved here. The proof of $(2)$ implies $(1)$ is quit easy from Definition 2.1 and [2, A.11 Thoerem]. Also, $(3)$ is equivalent to $(2)$ follows from the following isomorphism

[TABLE]

Assume that $M$ is $I$-Cohen Macaulay, Since, $\dim_{S}N/\mathfrak{m}N=0$, then

[TABLE]

[TABLE]

above equations hold because of [2, A.11 Thoerem]. Since $\dim_{R/I}M/IM=\dim_{R}M/IM$ and

[TABLE]

So, last equation can be written as:

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Then, Equations (2.1) and (2.2) imply that

[TABLE]

By [2, Proposition 1.1.2], $\operatorname{grade}(IS,M\otimes_{R}N)\geq\operatorname{grade}(I,M)$. This proves all the claims in $(2)$ and $(3)$. ∎

Corollary 2.8**.**

Let $f:R\to S$ be a ring map between Noetherian rings and $I$ any proper ideal of $R$. Suppose that $M$ and $N$ are finitely generated modules over $R$ and $S$ resp. If $N$ is flat over $R$, $I\subseteq\mathfrak{p}$ and $(M\otimes_{R}N)_{\mathfrak{p}}\neq I(M\otimes_{R}N)_{\mathfrak{p}}$, where $\mathfrak{p}=\mathfrak{q}\cap R$ and $\mathfrak{q}\in\operatorname{Spec}(S)$. Then the following conditions are equivalent:

• (1)

$M_{\mathfrak{p}}$* is $IR_{\mathfrak{p}}$-Cohen Macaulay and $\dim_{S_{\mathfrak{q}}}\frac{(M\otimes_{R}N)_{\mathfrak{p}}}{\mathfrak{p}(M\otimes_{R}N)_{\mathfrak{p}}}=0$.*

• (2)

$(M\otimes_{R}N)_{\mathfrak{p}}$* is $IS_{\mathfrak{q}}$-Cohen Macaulay, $\operatorname{grade}(IR_{\mathfrak{p}},M_{\mathfrak{p}})=\operatorname{grade}(IS_{\mathfrak{q}},(M\otimes_{R}N)_{\mathfrak{p}})$ and*

[TABLE]

• (3)

$(M\otimes_{R}N)_{\mathfrak{p}}$* is $IS_{\mathfrak{q}}$-Cohen Macaulay, $\operatorname{grade}(IR_{\mathfrak{p}},M_{\mathfrak{p}})=\operatorname{grade}(IS_{\mathfrak{q}},(M\otimes_{R}N)_{\mathfrak{p}})$ and*

[TABLE]

Proof.

It is a consequence of Theorem 2.7 in view of the fact that the ring map $R_{\mathfrak{p}}\to S_{\mathfrak{q}}$ is local. ∎

Proposition 2.9**.**

Let $I$ and $J$ be two proper ideals of $R$. Then

• (1)

If $M$ is $J$-Cohen Macaulay and $I\subseteq J$, then $M$ is $I$-Cohen Macaulay provided that $\operatorname{grade}(I,M)=\operatorname{grade}(J,M)$.

• (2)

If $M$ is $I$-Cohen Macaulay and $I\subseteq J$ , then $M$ is $J$-Cohen Macaulay provided that $\dim(M/JM)=\dim(M/IM)$.

• (3)

If $M$ is both $I$ and $J$-Cohen Macaulay, then $M$ is $I\cap J$-Cohen Macaulay.

Proof.

$(1)$ Since $\dim(\frac{M}{JM})\leq\dim(\frac{M}{IM})$ then,

[TABLE]

Hence, $M$ is $I$-Cohen Macaulay. With the similar arguments, $(2)$ can be proved. Also, $(3)$ follows from $(1)$ and [2, Proposition 1.2.10]. ∎

Cohen Macaulayness as well as almost Cohen Macaulayness are preserved when module is quotient by a regular sequence, see [2, Theorem 2.1.3] and [9, Lemma 2.7]. Same phenomena for $I$-Cohen Macaulay modules is given here.

Theorem 2.10**.**

Let $I$ be a proper ideal of a Noetherian local ring $R$, $M$ an $R$-module and $x_{1},x_{2},\cdots x_{r}$ an $M$-regular sequence in $I$. Then $M$ is $I$-Cohen Macaulay if and only if $\frac{M}{<x_{1},x_{2},\cdots x_{r}>M}$ is $\frac{I}{<x_{1},x_{2},\cdots x_{r}>}$-Cohen Macaulay over $\frac{R}{<x_{1},x_{2},\cdots x_{r}>}$.

Proof.

Note that there is an isomorphism

[TABLE]

where $\acute{M}=\frac{M}{<x_{1},x_{2},\cdots x_{r}>M}$ and $\acute{I}=\frac{I}{<x_{1},x_{2},\cdots x_{r}>}$. So, the proof follows directly from Definition 2.1, Equation (2.1) and from the following equality between grades:

[TABLE]

(see [2, Proposition 1.2.10(d)]). ∎

For any $\mathfrak{p}\in\operatorname{Ass}_{R}(M)$, a relation between dimension of a Cohen Macaulay module and dimension of $R/\mathfrak{p}$ is proved in [2, Theorem 2.1.2]. $i.e.$ a Cohen Macaulay module has no embedded primes. A similar relation is developed for $\mathfrak{p}$-Cohen Macaulay modules in the following result.

Theorem 2.11**.**

Let $M$ be a $\mathfrak{p}$-Cohen Macaulay module over a local ring $R$, where $\mathfrak{p}\in\operatorname{Supp}_{R}(M)$. Then

[TABLE]

Proof.

If $\operatorname{grade}(\mathfrak{p},M)=0$, then $\mathfrak{q}\subseteq\mathfrak{p}$ for some $\mathfrak{q}\in\operatorname{Ass}_{R}(M)$. Since $\dim_{R}M/\mathfrak{p}M\leq\dim_{R}R/\mathfrak{p}$ and M is $\mathfrak{p}$-Cohen Macaulay, Then

[TABLE]

Now, if $\operatorname{grade}(\mathfrak{p},M)>0$. Then, there exists an $M$-regular element $x\in\mathfrak{p}$. Let $\bar{M}=M/xM$, then, by Theorem 2.10, $\bar{M}$ is $\mathfrak{p}/<x>$-Cohen Macaulay. By induction on $\bar{M}$,

[TABLE]

[TABLE]

By [11, 9.A Proposition], $\mathfrak{q}_{1}\in\operatorname{Ass}_{R}(\bar{M})$ such that $\mathfrak{q}_{1}\subseteq\mathfrak{p}$. Then $\mathfrak{q}_{1}\in\operatorname{Supp}_{R}(\bar{M})$. In particular, $\mathfrak{q}_{1}\in\operatorname{Supp}_{R}(M)$ and $x\in\mathfrak{q}_{1}$. Then, there exists $\mathfrak{q}\in\operatorname{Ass}_{R}(M)$ such that $\mathfrak{q}\subsetneq\mathfrak{q}_{1}\subseteq\mathfrak{p}$. Hence

[TABLE]

This proves that $\dim_{R}R/\mathfrak{q}=\dim_{R}M$, where $\mathfrak{q}\in\operatorname{Ass}_{R}(M)$ such that $\mathfrak{q}\subseteq\mathfrak{p}$. ∎

Lemma 2.12**.**

With the same assumptions as in Theorem 2.10, assume that $S$ is a multiplicative closed subset of $R$ and $\mathfrak{p}\in\operatorname{Supp}_{R}(M)$ such that $\mathfrak{p}\cap S=\emptyset$. If $M$ is $\mathfrak{p}$-Cohen Macaulay, then

• (1)

$\operatorname{grade}(\mathfrak{p},M)=\operatorname{grade}(S^{-1}\mathfrak{p},S^{-1}M).$**

• (2)

If $\dim_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\geq 1$. Then, $\operatorname{grade}(\mathfrak{p},M)\geq 1$.

Proof.

$(1)$ If $\operatorname{grade}(\mathfrak{p},M)=0$. Form the proof of Theorem 2.11, it follows that $\mathfrak{p}\in\operatorname{Ass}_{R}(M)$. Hence, $S^{-1}\mathfrak{p}\in\operatorname{Ass}_{S^{-1}R}(S^{-1}M)$ and $\operatorname{grade}(\mathfrak{p},M)=\operatorname{grade}(S^{-1}\mathfrak{p},S^{-1}M)=0$.

If $\operatorname{grade}(\mathfrak{p},M)>0$. Then, there exists an $M$-regular element $x\in\mathfrak{p}$. Note that $x\in S^{-1}\mathfrak{p}$ is also $S^{-1}M$-regular. Let $\bar{M}=M/xM$. By induction on $\bar{M}$

[TABLE]

So, the result is proved by [2, Proposition 1.2.10].

$(2)$ It is easy in view of arguments presented above. ∎

Problem 2.13*.*

Perseverance of $I$-Cohen Macaulayness in module of fractions is still an open problem. However, in case of localization at a prime ideal is discussed in Theorem 2.14.

By [2, Theorem 2.1.7], localization at prime ideals of a Cohen Macaulay module is Cohen Macaulay. Same is true (with some conditions on ideal) for almost Cohen Macaulay modules, see [2, Lemma 2.6]. Further, system of parameters and maximal regular sequences coincide in Cohen Macaulay modules and their relation in almost Cohen Macaulay modules is described in [9, Theorem 1.7]. In case of $I$-Cohen Macaulay modules these results are developed in next Theorem.

Theorem 2.14**.**

With the previous notion, let $M$ be a $\mathfrak{p}$-Cohen Macaulay. Then the following statements hold:

• (1)

$M_{\mathfrak{p}}$* is Cohen Macaulay over $R_{\mathfrak{p}}$.*

• (2)

Ever maximal $M$-regular sequence in $\mathfrak{p}$ is a system of parameters of $M_{\mathfrak{p}}$. That is if $x_{1},\dots,x_{n}\in\mathfrak{p}$ is a maximal regular sequence over $M$. Then $\frac{x_{1}}{1},\dots,\frac{x_{n}}{1}\in\mathfrak{p}R_{\mathfrak{p}}$ is a system of parameters of $M_{\mathfrak{p}}$.

• (3)

For any $\mathfrak{q}\in\operatorname{Supp}_{R}(M)$ such that $\mathfrak{q}\subseteq\mathfrak{p}$, $M_{\mathfrak{q}}$ is Cohen Macaulay over $R_{\mathfrak{q}}$.

Proof.

Note that $(2)$ and $(3)$ are obvious in view of $(1)$ and Lemma 2.12. It is remain to prove the claim in $(1)$. If $\dim_{R}M=0$, then the result is true. So, let $\dim_{R}M\geq 1$, then by Lemma 2.12, $\operatorname{grade}(\mathfrak{p},M)\geq 1.$ So, there exists $x\in\mathfrak{p}$ which is $M$-regular. Also, it is $M_{\mathfrak{p}}$-regular in $\mathfrak{p}R_{\mathfrak{p}}$. Let $\bar{M}=M/xM$, then $\bar{M}$ is $\mathfrak{p}/x$-Cohen Macaulay, see Theorem 2.10. By induction, $\bar{M}_{\mathfrak{p}}$ is Cohen Macaulay over $\bar{R}_{\mathfrak{p}}$. Hence $M_{\mathfrak{p}}/xM_{\mathfrak{p}}$ is Cohen Macaulay over $R_{\mathfrak{p}}/xR_{\mathfrak{p}}$. Now assertion follows from [11, Proposition 19.3.3]. ∎

Remark 2.15*.*

By Example 2.3$(1)$, the statements of Corollary 2.14 are not true, if $R$ is not local.

Since, the polynomial ring $R[x_{1},\dots,x_{n}]$ and formal power series ring $R[|x_{1},\dots,x_{n}|]$ inherit both Cohen Macaulayness and almost Cohen Macaulayness properties of $R$, see [2, Theorem 2.1.9] and [9, Theorems 1.3 and 1.6]. Same is true for $I$-Cohen Macaulay modules.

Theorem 2.16**.**

Let $I$ be a proper ideal of $R$. Then the following condition are equivalent:

• (1)

$R$* is $I$-Cohen Macaulay*

• (2)

$R[x_{1},\dots,x_{n}]$* is $I[x_{1},\dots,x_{n}]$-Cohen Macaulay and*

[TABLE]

• (3)

$R[|x_{1},\dots,x_{n}|]$* is $I[|x_{1},\dots,x_{n}|]$-Cohen Macaulay and*

[TABLE]

Proof.

It is enough to prove the result for $n=1$. Suppose that $(1)$ is true. Since $R\to R[x]$ is a flat ring map, then

[TABLE]

This proves $(2)$. The implication $(2)\Rightarrow(1)$ is straight forward. The proof of $(1)$ is equivalent to $(3)$ is so similar. ∎

Corollary 2.17**.**

Let $I$ be any proper ideal of $R$ and $\mathfrak{p}\in\operatorname{Spec}(R)$ such that $\operatorname{grade}(I,R)=0pt(I)$ and $\operatorname{grade}(\mathfrak{p},R)=0pt(\mathfrak{p})$. Then the following conditions hold:

• (1)

$R$* is $I$-Cohen Macaulay if and only if $R[|x_{1},\dots,x_{n}|]$ is $I[|x_{1},\dots,x_{n}|]$-Cohen Macaulay.*

• (2)

$R$* is $\mathfrak{p}$-Cohen Macaulay if and only if $R[x_{1},\dots,x_{n}]$ is $\mathfrak{p}[x_{1},\dots,x_{n}]$-Cohen Macaulay.*

Proof.

Since,$\phi:R\to R[|x|]$ is a faithfully flat ring map, see [1, Exercise 10.12]. By Theorem 2.16, the result is obvious in view of the following identities:

[TABLE]

see [11, (13.B)-Thoerem 19] and [5, Exercise 18] ∎

Let $I$ be an ideal of $R$ contained in the Jacobson radical of $R$. Let $\hat{R}$ denote the $I$-adic completion of $R$. Then, [2, Corollary 2.1.8] and [7, Corollary 2.4] state that $R$ is Cohen Macaulay ( resp. almost Cohen Macaulay) if and only if $\hat{R}$ is Cohen Macaulay ( resp. almost Cohen Macaulay).

Theorem 2.18**.**

Assuming above notions, $R$ is $I$-Cohen Macaulay if and only if $\hat{R}$ is $I\hat{R}$-Cohen Macaulay and $\operatorname{grade}(I,R)=\operatorname{grade}(I\hat{R},\hat{R})$.

Proof.

The proof goes on the same lines to the proof of Theorem 2.16 with following information:

[TABLE]

[TABLE]

∎