Cohomological dimension and relative Cohen-Maculayness

TL;DR

This paper introduces a-relative system of parameters in commutative algebra, characterizes them through cohomological dimension, and provides a criterion for relative Cohen-Macaulay modules, advancing the understanding of these structures.

Contribution

It defines a-relative system of parameters, links them to cohomological dimension, and establishes a new criterion for identifying relative Cohen-Macaulay modules.

Findings

Introduction of a-relative system of parameters

Characterization of these systems via cohomological dimension

Criterion for relative Cohen-Macaulay modules

Abstract

Let R be a commutative Noetherian (not necessarily local) ring with identity and a be a proper ideal of R. We introduce a notion of a-relative system of parameters and characterize them by using the notion of cohomological dimension. Also, we present a criterion of relative Cohen-Macaulay modules via relative system of parameters.

Equations

$$\mathrm{cd}\left(\mathfrak{a},M/⟨x_1,x_2,\dots ,x_i⟩M\right)=c-i$$

$$\mathrm{Rad}\left(⟨x_1,x_2,\dots ,x_d⟩+{\mathrm{Ann}}_{R}M\right)=\mathrm{Rad}\left(\mathfrak{m}+{\mathrm{Ann}}_{R}M\right).$$

$$\mathrm{Rad}\left(⟨x_1,x_2,\dots ,x_c⟩+{\mathrm{Ann}}_{R}M\right)=\mathrm{Rad}\left(\mathfrak{a}+{\mathrm{Ann}}_{R}M\right).$$

$$\mathrm{cd}\left(\mathfrak{a},M/⟨x_1,x_2,\dots ,x_i⟩M\right)=c-i$$

$${\text{H}}_{\mathfrak{a}}^i\left(M\right):=\underset{n\in \mathbb{N}}{\underrightarrow{\lim }}{\text{Ext}}_R^i\left(R/{\mathfrak{a}}^n,M\right);i\in {\mathbb{N}}_0.$$

$$\mathrm{Rad}\left(⟨x_1,x_2,\dots ,x_c⟩+{\mathrm{Ann}}_{R}M\right)=\mathrm{Rad}\left(\mathfrak{a}+{\mathrm{Ann}}_{R}M\right).$$

$$\mathrm{Rad}\left(⟨x_1,x_2,\dots ,x_n⟩+{\mathrm{Ann}}_{R}M\right)=\mathrm{Rad}\left(\mathfrak{a}+{\mathrm{Ann}}_{R}M\right).$$

$$\mathrm{Rad}\left(⟨x_1,x_2,\dots ,x_c⟩+{\mathrm{Ann}}_{R}M\right)=\mathrm{Rad}\left(\mathfrak{a}+{\mathrm{Ann}}_{R}M\right),$$

$$\mathrm{Rad}\left(⟨y_1,\dots ,y_c⟩+{\mathrm{Ann}}_{R}M\right)=\mathrm{Rad}\left(\mathfrak{a}+{\mathrm{Ann}}_{R}M\right).$$

$$\mathrm{Rad}\left(⟨z_1,z_2,\dots ,z_c⟩+{\mathrm{Ann}}_{R}M\right)=\mathrm{Rad}\left(\mathfrak{a}+{\mathrm{Ann}}_{R}M\right),$$

$$U:=\left\{\left(x_1^{{\alpha }_1},\dots ,x_n^{{\alpha }_n}\right)|{\alpha }_1,\dots ,{\alpha }_n\in \mathbb{N}\right\}.$$

$$\left(r,\left(x_1^{{\alpha }_1},\dots ,x_n^{{\alpha }_n}\right)\right)\sim \left(s,\left(x_1^{{\beta }_1},\dots ,x_n^{{\beta }_n}\right)\right)$$

$$x_1^{{\delta }_1-{\alpha }_1}\dots x_n^{{\delta }_n-{\alpha }_n}r-x_1^{{\delta }_1-{\beta }_1}\dots x_n^{{\delta }_n-{\beta }_n}s\in ⟨x_1^{{\delta }_1},\dots ,x_{n-1}^{{\delta }_{n-1}}⟩M.$$

$${\text{H}}_{\mathfrak{a}}^{d-i}\left(\frac{M}{⟨x_1,\dots ,x_{i-1},x_i⟩M}\right)\to {\text{H}}_{\mathfrak{a}}^{d-i+1}\left(\frac{M}{⟨x_1,\dots ,x_{i-1}⟩M}\right)\stackrel{x_i}{⟶}{\text{H}}_{\mathfrak{a}}^{d-i+1}\left(\frac{M}{⟨x_1,\dots ,x_{i-1}⟩M}\right)\to 0.$$

$${\text{H}}_{\mathfrak{a}}^{d-1}\left(M/x_{1}M\right)⟶{\text{H}}_{\mathfrak{a}}^d\left(M\right)\stackrel{x_1}{⟶}{\text{H}}_{\mathfrak{a}}^d\left(M\right)⟶0.$$

$$U:=\left\{\left(x_1^{{\alpha }_1},x_2^{{\alpha }_2},\dots ,x_d^{{\alpha }_d},1\right)|{\alpha }_1,\dots ,{\alpha }_d\in \mathbb{N}\right\}$$

$$V:=\left\{\left(x_2^{{\alpha }_2},x_3^{{\alpha }_3},\dots ,x_d^{{\alpha }_d},1\right)|{\alpha }_2,\dots ,{\alpha }_d\in \mathbb{N}\right\}.$$

$$\phi :V^{-d}\overline{M}⟶U^{-d-1}M$$

$$\phi \left(\frac{\overline{r}}{\left(x_2^{{\alpha }_2},x_3^{{\alpha }_3},\dots ,x_d^{{\alpha }_d},1\right)}\right)=\frac{r}{\left(x_1,x_2^{{\alpha }_2},\dots ,x_d^{{\alpha }_d},1\right)}$$

$$V^{-d}\overline{M}\stackrel{\phi }{⟶}{U}^{-d-1}M\stackrel{\psi }{⟶}{U}^{-d-1}M⟶0$$

$$x_1^{\delta +1-{\alpha }_1}{x}_2^{\delta -{\alpha }_2}\dots x_d^{\delta -{\alpha }_d}r=\sum _{i=1}^d{x}_i^{\delta }{r}_i,$$

$$\begin{array}[]{ll}z&=\frac{r}{\left(x_{1}^{\alpha_{1}},\ldots,x_{d}^{\alpha_{d}},1\right)}\\ &=\frac{x_{1}^{\delta+1-\alpha_{1}}x_{2}^{\delta-\alpha_{2}}\ldots x_{d}^{\delta-\alpha_{d}}r}{\left(x_{1}^{\delta+1},x_{2}^{\delta},\ldots,x_{d}^{\delta},1\right)}\\ &=\underset{i=1}{\overset{d}{\sum}}\frac{x_{i}^{\delta}r_{i}}{\left(x_{1}^{\delta+1},x_{2}^{\delta},\ldots,x_{d}^{\delta},1\right)}\\ &=\frac{x_{1}^{\delta}r_{1}}{\left(x_{1}^{\delta+1},x_{2}^{\delta},\ldots,x_{d}^{\delta},1\right)}\\ &=\frac{r_{1}}{\left(x_{1},x_{2}^{\delta},\ldots,x_{d}^{\delta},1\right)}\\ &=\varphi\left(\frac{\overline{r}_{1}}{\left(x_{2}^{\delta},x_{3}^{\delta},\ldots,x_{d}^{\delta},1\right)}\right).\end{array}$$

$${\text{H}}_{\mathfrak{b}}^{d-i}\left(N/x_{i}N\right)⟶{\text{H}}_{\mathfrak{b}}^{d-i+1}\left(N\right)\stackrel{x_i}{⟶}{\text{H}}_{\mathfrak{b}}^{d-i+1}\left(N\right)⟶0,$$

$$\mathrm{Rad}\left(\mathfrak{a}+{\mathrm{Ann}}_{R}M\right)=\mathrm{Rad}\left(⟨x_1,x_2,\dots ,x_c⟩+{\mathrm{Ann}}_{R}M\right),$$

$${\text{H}}_{\mathfrak{a}}^{c-i}\left(\overline{M}/x_i\overline{M}\right)\to {\text{H}}_{\mathfrak{a}}^{c-i+1}\left(\overline{M}\right)\stackrel{x_i}{⟶}{\text{H}}_{\mathfrak{a}}^{c-i+1}\left(\overline{M}\right)\to 0.$$

$${\text{H}}_{\mathfrak{a}}^{c-i+1}\left(\overline{M}\right)\stackrel{x_i}{⟶}{\text{H}}_{\mathfrak{a}}^{c-i+1}\left(\overline{M}\right)$$

$$c-i\le \mathrm{cd}\left(\mathfrak{a},\overline{M}/x_i\overline{M}\right)=\mathrm{cd}\left(\mathfrak{a},M/⟨x_1,x_2,\dots ,x_i⟩M\right).$$

$$\begin{array}[]{ll}\operatorname{Rad}\left(\mathfrak{a}+\operatorname{Ann}_{R}\left(M/\langle x_{1},x_{2},\ldots,x_{i}\rangle M\right)\right)&=\operatorname{Rad}\left(\mathfrak{a}+\left(\langle x_{1},x_{2},\ldots,x_{i}\rangle+\operatorname{Ann}_{R}M\right)\right)\\ &=\operatorname{Rad}\left(\mathfrak{a}+\operatorname{Ann}_{R}M\right)\\ &=\operatorname{Rad}\left(\langle x_{1},x_{2},\ldots,x_{c}\rangle+\operatorname{Ann}_{R}M\right)\\ &=\operatorname{Rad}\left(\langle x_{i+1},\ldots,x_{c}\rangle+\left(\langle x_{1},x_{2},\ldots,x_{i}\rangle+\operatorname{Ann}_{R}M\right)\right)\\ &=\operatorname{Rad}\left(\langle x_{i+1},\ldots,x_{c}\rangle+\operatorname{Ann}_{R}\left(M/\langle x_{1},x_{2},\ldots,x_{i}\rangle M\right)\right).\end{array}$$

$$\mathrm{ara}\left(\mathfrak{a},M/⟨x_1,x_2,\dots ,x_i⟩M\right)=\mathrm{cd}\left(\mathfrak{a},M/⟨x_1,x_2,\dots ,x_i⟩M\right)=c-i,$$

$$\mathrm{Rad}\left(\mathfrak{a}+{\mathrm{Ann}}_{R}M\right)=\mathrm{Rad}\left(⟨y⟩+{\mathrm{Ann}}_{R}M\right).$$

Full text

1. Introduction

Throughout, the word ring stands for commutative Noetherian rings with identity. Consider the following naturally-raised questions:

Question 1.1**.**

Let $\mathfrak{a}$ be a proper ideal of a ring $R$, $M$ a finitely generated $R$-module and $c=\operatorname{cd}(\mathfrak{a},M)$. Is there a sequence $x_{1},x_{2},\ldots,x_{c}$ of elements in $\mathfrak{a}$ such that

[TABLE]

for every $i=1,2,\ldots,c$? If yes, how can we characterize such sequences?

Let $R$ be a ring, $\mathfrak{a}$ an ideal of $R$ and $M$ a finitely generated $R$-module with $M\neq\mathfrak{a}M$. Then $M$ is said to be $\mathfrak{a}$-relative Cohen-Macaulay, $\mathfrak{a}$-RCM, if $\operatorname{grade}(\mathfrak{a},M)=\operatorname{cd}(\mathfrak{a},M)$. This notion was introduced by Majid Rahro Zargar and the fourth author in [RZ2] and its study was continued in [Ra1], [Ra2], [Ra3], [RZ1] and [CH]. Relative Cohen-Macaulay bigraded modules were already introduced and investigated by Ahad Rahimi; see [R] and [JR]. Also, the closely related notion of cohomologically complete intersection ideals was examined by Michael Hellus and Peter Schenzel in [HS].

Question 1.2**.**

Over a local ring $T,$ a finitely generated $T$-module $N$ is Cohen-Macaulay if and only if every system of parameters of $N$ is an $N$-regular sequence. Is there an analogue characterization for $\mathfrak{a}$-relative Cohen-Macaulay $R$-modules?

This paper is dealing with the above questions. Although these questions don’t look so related in the beginning, surprisingly, they become connected through a notion of relative system of parameters. Here, we introduce this notion and through investigation of its properties, we answer the above questions.

Let $c:=\operatorname{cd}(\mathfrak{a},M)$ denote the cohomological dimension of $M$ with respect to $\mathfrak{a}$; i.e. the supermum of the integers $i$ for which $\text{H}_{\mathfrak{a}}^{i}(M)\neq 0$. Recall that when $R$ is local with the unique maximal ideal $\mathfrak{m}$ and $\dim_{R}M=d$, a sequence $x_{1},x_{2},\ldots,x_{d}\in\mathfrak{m}$ is called a system of parameters of $M$ if the $R$-module $M/\langle x_{1},x_{2},\ldots,x_{d}\rangle M$ has finite length. This is equivalent to say that

[TABLE]

We call a sequence $x_{1},x_{2},\ldots,x_{c}\in\mathfrak{a}$ an $\mathfrak{a}$-relative system of parameters, $\mathfrak{a}$-Rs.o.p, of $M$ if

[TABLE]

System of parameters appear in many contexts. Especially, Monomial Conjecture on system of parameters of local rings stands for decades until recently solved by Yves Andr$\acute{e}$; see [An]. Although over a local ring every finitely generated $R$-module possesses a system of parameters, this is not the case for $\mathfrak{a}$-relative systems of parameters. It is immediate that $R$ admits an $\mathfrak{a}$-relative system of parameters if and only if $\operatorname{ara}(\mathfrak{a})=\operatorname{cd}(\mathfrak{a},R)$. Let $K$ be a field. For a square-free monomial ideal $\mathfrak{a}$ of a polynomial ring $R=K[x_{1},\dots,x_{n}]$, it is known that $\operatorname{cd}(\mathfrak{a},R)=\operatorname{pd}_{R}\frac{R}{\mathfrak{a}}$; see [Ly, Theorem 1]. Characterizing monomial ideals $\mathfrak{a}$ satisfying $\operatorname{ara}(\mathfrak{a})=\operatorname{pd}_{R}\frac{R}{\mathfrak{a}}$ has been an active area of research for years; see e.g. [Ba1], [Ba2] and [SV].

Assume that $\mathfrak{a}$ is contained in the Jacobson radical of $R$ and $M$ possesses an $\mathfrak{a}$-Rs.o.p. We prove that a sequence $\underline{x}=x_{1},x_{2},\ldots,x_{c}\in\mathfrak{a}$ is $\mathfrak{a}$-relative system of parameters of $M$ if and only if

[TABLE]

for every $1\leq i\leq c$; see Theorem 2.7. Also, we show that $M$ is $\mathfrak{a}$-relative Cohen-Macaulay if and only if every $\mathfrak{a}$-relative system of parameters of $M$ is an $M$-regular sequence if and only if there exists an $\mathfrak{a}$-relative system of parameters of $M$ which is an $M$-regular sequence; see Theorem 3.3. These two results yields that if $M$ is $\mathfrak{a}$-RCM and $\underline{x}=x_{1},x_{2},\ldots,x_{c}\in\mathfrak{a}$ is an $\mathfrak{a}$-Rs.o.p of $M$, then $M/\langle x_{1},x_{2},\ldots,x_{i}\rangle M$ is $\mathfrak{a}$-RCM for every $i=1,\ldots,c$; see Corollary 3.5.

2. Question 1.1

Theorem 2.7 is the main result of this paper. To prove it, we need Lemmas 2.3, 2.4, 2.5 and 2.6. We begin by recalling some needed definitions.

Let $\mathfrak{a}$ be an ideal of $R$ and $M$ a finitely generated $R$-module. Recall that the arithmetic rank of $\mathfrak{a}$, denoted by $\operatorname{ara}\left(\mathfrak{a}\right)$, is the least number of elements of $R$ required to generate an ideal with the same radical as $\mathfrak{a}$. Among other things, this paper deals with the local cohomology modules

[TABLE]

If $\mathfrak{b}$ is another ideal of $R$ such that the ideals $\mathfrak{a}+\operatorname{Ann}_{R}M$ and $\mathfrak{b}+\operatorname{Ann}_{R}M$ have the same radical, then the Independence Theorem [BS, Theorem 4.2.1] yields a natural $R$-isomorphism $\text{H}_{\mathfrak{a}}^{i}\left(M\right)\cong\text{H}_{\mathfrak{b}}^{i}\left(M\right)$ for all $i\in\mathbb{N}_{0}.$ One easily sees that $\operatorname{cd}\left(\mathfrak{a},M\right)=-\infty$ if and only if $M=\mathfrak{a}M$. On the other hand, [BS, Corollary 3.3.3] implies that $\operatorname{cd}\left(\mathfrak{a},M\right)\leq\operatorname{ara}\left(\mathfrak{a}\right)$. In the case $\left(R,\mathfrak{m}\right)$ is a local ring, it is known that $\operatorname{ara}\left(\mathfrak{m}\right)=\dim R=\operatorname{cd}\left(\mathfrak{m},R\right)$.

Definition 2.1**.**

Let $M$ be a finitely generated $R$-module and $\mathfrak{a}$ an ideal of $R$ with $M\neq\mathfrak{a}M$.

1. i)

Let $c=\operatorname{cd}\left(\mathfrak{a},M\right)$. A sequence $x_{1},x_{2},\ldots,x_{c}\in\mathfrak{a}$ is called $\mathfrak{a}$-relative system of parameters, $\mathfrak{a}$-Rs.o.p, of $M$ if

[TABLE] 2. ii)

Arithmetic rank of $\mathfrak{a}$ with respect to $M$, $\operatorname{ara}\left(\mathfrak{a},M\right)$, is defined as the infimum of the integers $n\in\mathbb{N}_{0}$ such that there exist $x_{1},x_{2},\ldots,x_{n}\in R$ satisfying

[TABLE]

Clearly if $x_{1},x_{2},\ldots,x_{c}\in R$ is an $\mathfrak{a}$-Rs.o.p of $M$, then for all $t_{1},\ldots,t_{c}\in\mathbb{N}$, every permutation of $x_{1}^{t_{1}},\ldots,x_{c}^{t_{c}}$ is also an $\mathfrak{a}$-Rs.o.p of $M$. One may easily check that $\operatorname{cd}\left(\mathfrak{a},M\right)\leq\operatorname{ara}\left(\mathfrak{a},M\right)$. Obviously, $\operatorname{ara}\left(\mathfrak{a},R\right)=\operatorname{ara}\left(\mathfrak{a}\right)$.

Our first result provides a characterization for existence of relative system of parameters. Although it is an easy observation, we include its proof for the reader’s convenience.

Lemma 2.2**.**

Let $M$ be a finitely generated $R$-module and $\mathfrak{a}$ an ideal of $R$ with $M\neq\mathfrak{a}M$. Then $\mathfrak{a}$ contains an $\mathfrak{a}$-Rs.o.p of $M$ if and only if $\operatorname{ara}\left(\mathfrak{a},M\right)=\operatorname{cd}\left(\mathfrak{a},M\right)$.

Proof.

Set $c:=\operatorname{cd}\left(\mathfrak{a},M\right)$. Let $x_{1},x_{2},\ldots,x_{c}\in\mathfrak{a}$ be an $\mathfrak{a}$-Rs.o.p of $M$. Then

[TABLE]

and so $\operatorname{ara}\left(\mathfrak{a},M\right)\leq c\leq\operatorname{ara}\left(\mathfrak{a},M\right)$. Thus $\operatorname{ara}\left(\mathfrak{a},M\right)=c$.

Next, suppose that $\operatorname{ara}\left(\mathfrak{a},M\right)=\operatorname{cd}\left(\mathfrak{a},M\right)$. Hence, there are $y_{1},y_{2},\ldots,y_{c}$ in $R$ such that

[TABLE]

There is $n\in\mathbb{N}$ such that $y_{i}^{n}\in\mathfrak{a}+\operatorname{Ann}_{R}M$ for every $1\leq i\leq c$. So for each $1\leq i\leq c$, there are $z_{i}\in\mathfrak{a}$ and $w_{i}\in\operatorname{Ann}_{R}M$ such that $y_{i}^{n}=z_{i}+w_{i}$. Now,

[TABLE]

and so $z_{1},z_{2},\ldots,z_{c}\in\mathfrak{a}$ is an $\mathfrak{a}$-Rs.o.p of $M$. ∎

This note is also concerned with the special case of the notion of generalized fractions. This notion is described as follows: Let $x_{1},\ldots,x_{n}$ be a sequence of elements of $R$ and $M$ an $R$-module. Set

[TABLE]

Then $U$ leads to a module of generalized fractions $U^{-n}M$:

For every $r,s\in M$ and $\left(x_{1}^{\alpha_{1}},\ldots,x_{n}^{\alpha_{n}}\right),\left(x_{1}^{\beta_{1}},\ldots,x_{n}^{\beta_{n}}\right)\in U$, we write

[TABLE]

if there exist integers $\delta_{i}\geq\max\{\alpha_{i},\beta_{i}\}$; $i=1,\ldots,n$ such that

[TABLE]

It is easy to verify that $\sim$ is an equivalence relation on $M\times U$. Then the equivalence class of an element $\left(r,\left(x_{1}^{\alpha_{1}},\ldots,x_{n}^{\alpha_{n}}\right)\right)$ is denoted by $\frac{r}{\left(x_{1}^{\alpha_{1}},\ldots,x_{n}^{\alpha_{n}}\right)}$ and we let $U^{-n}M$ stand for the set of all equivalence classes of $\sim$. With naturally defined sum and scalar multiplication, $U^{-n}M$ forms an $R$-module. It is easy to see that $\frac{r}{\left(x_{1}^{\alpha_{1}},\ldots,x_{n}^{\alpha_{n}}\right)}\in U^{-n}M$ is zero if and only if there exists an integer $\delta\geq\max\{\alpha_{1},\ldots,\alpha_{n}\}$ such that $x_{1}^{\delta-\alpha_{1}}\ldots x_{n}^{\delta-\alpha_{n}}r\in\langle x_{1}^{\delta},\ldots,x_{n-1}^{\delta}\rangle M$. For more details see [SZ].

The next result is very crucial in this paper and may also have applications in other contexts.

Lemma 2.3**.**

Let $\mathfrak{a}=\langle x_{1},\ldots,x_{d}\rangle$ be an ideal of $R$ and $M$ a finitely generated $R$-module. Then for every $i=1,\ldots,d,$ one has the following exact sequence

[TABLE]

In particular, there is an exact sequence

[TABLE]

Proof.

We first prove the last assertion. Denote $M/x_{1}M$ by $\overline{M}$ and let $-:M\longrightarrow\overline{M}$ be the natural epimorphism. Set

[TABLE]

and

[TABLE]

Then, by [KSZ, Remark 2.2], $\text{H}_{\mathfrak{a}}^{d}\left(M\right)\cong U^{-d-1}M$ and $\text{H}_{\mathfrak{a}}^{d-1}\left(\overline{M}\right)\cong V^{-d}\overline{M}$. Define

[TABLE]

by

[TABLE]

and let $\psi:U^{-d-1}M\longrightarrow U^{-d-1}M$ denote the map defined by multiplication by $x_{1}$. It suffices to show that the sequence

[TABLE]

is exact. Let $z=\frac{r}{\left(x_{1}^{\alpha_{1}},\ldots,x_{d}^{\alpha_{d}},1\right)}\in U^{-d-1}M$. Then $z=\frac{x_{1}r}{\left(x_{1}^{\alpha_{1}+1},x_{2}^{\alpha_{2}},\ldots,x_{d}^{\alpha_{d}},1\right)}$, and so $\psi$ is surjective.

Clearly, $\operatorname{im}\varphi\subseteq\ker\psi$. Now, let $\psi\left(z\right)=0$. Then, there is an integer $\delta\geq\max\{\alpha_{1},\ldots,\alpha_{d}\}$ such that $x_{1}^{\delta-\alpha_{1}}\ldots x_{d}^{\delta-\alpha_{d}}x_{1}r\in\langle x_{1}^{\delta},\ldots,x_{d}^{\delta}\rangle M$. Hence,

[TABLE]

where $r_{1},\ldots,r_{d}\in M$. This yields that

[TABLE]

So, $\ker\psi\subseteq\operatorname{im}\varphi$. This completes the proof of the last assertion.

Now, we show the first assertion. Set $N:=M/\langle x_{1},...,x_{i-1}\rangle M$. Then by using the same argument as above, we have the following exact sequence

[TABLE]

where $\mathfrak{b}:=\langle x_{i},x_{i+1},\ldots,x_{d}\rangle$. This yields our claim. ∎

Lemma 2.4**.**

Let $M$ be a finitely generated $R$-module and $\mathfrak{a}$ an ideal of $R$ with $M\neq\mathfrak{a}M$. Let $c:=\operatorname{cd}\left(\mathfrak{a},M\right)$ and $\underline{x}=x_{1},x_{2},\ldots,x_{c}\in\mathfrak{a}$ be an $\mathfrak{a}$-Rs.o.p of $M$. Then for every $0\leq i\leq c$, the sequence $x_{i+1},x_{i+2},\ldots,x_{c}$ forms an $\mathfrak{a}$-Rs.o.p of $M/\langle x_{1},x_{2},\ldots,x_{i}\rangle M$, and so $\operatorname{cd}\left(\mathfrak{a},M/\langle x_{1},x_{2},\ldots,x_{i}\rangle M\right)=c-i$.

Proof.

We do induction on $i$. The case $i=0$ holds trivially. Next, assume that $i>0$ and the claim holds for $i-1$. Set $\overline{M}:=M/\langle x_{1},x_{2},\ldots,x_{i-1}\rangle M$. Then, by the induction hypothesis, $\operatorname{cd}\left(\mathfrak{a},\overline{M}\right)=c-i+1$. As

[TABLE]

Lemma 2.3 yields the exact sequence:

[TABLE]

Since $\text{H}_{\mathfrak{a}}^{c-i+1}\left(\overline{M}\right)$ is $\mathfrak{a}$-torsion and $x_{i}\in\mathfrak{a}$, each element of $\text{H}_{\mathfrak{a}}^{c-i+1}\left(\overline{M}\right)$ is annihilated by some power of $x_{i}$. Hence, as $\text{H}_{\mathfrak{a}}^{c-i+1}\left(\overline{M}\right)$ is nonzero, the map

[TABLE]

is not injective. So, $\text{H}_{\mathfrak{a}}^{c-i}\left(\overline{M}/x_{i}\overline{M}\right)\neq 0$. Consequently,

[TABLE]

Now, one has the following display of equalities:

[TABLE]

So, $\operatorname{ara}\left(\mathfrak{a},M/\langle x_{1},x_{2},\ldots,x_{i}\rangle M\right)\leq c-i.$ Thus

[TABLE]

and the sequence $x_{i+1},x_{i+2},\ldots,x_{c}$ is an $\mathfrak{a}$-Rs.o.p of $M/\langle x_{1},x_{2},\ldots,x_{i}\rangle M$. ∎

Let $\mathfrak{a}$ be an ideal of $R$ and $M,N$ two finitely generated $R$-modules such that $\operatorname{Supp}_{R}N\subseteq\operatorname{Supp}_{R}M$. Then, by [DNT, Theorem 2.2], $\operatorname{cd}(\mathfrak{a},N)\leq\operatorname{cd}(\mathfrak{a},M)$. In particular if $\operatorname{Supp}_{R}N=\operatorname{Supp}_{R}M$, then $\operatorname{cd}(\mathfrak{a},N)=\operatorname{cd}(\mathfrak{a},M)$. In the rest of the paper, we shall use this several times without any further comment.

Lemma 2.5**.**

Let $\mathfrak{a}$ be an ideal of $R$ which is contained in the Jacobson radical of $R$ and $x$ an element of $\mathfrak{a}$. Assume that $M$ is a nonzero finitely generated $R$-module with $\operatorname{ara}\left(\mathfrak{a},M\right)=1$. If $\text{H}_{\mathfrak{a}}^{1}\left(M/xM\right)=0$, then $\operatorname{Rad}\left(\mathfrak{a}+\operatorname{Ann}_{R}M\right)=\operatorname{Rad}\left(\langle x\rangle+\operatorname{Ann}_{R}M\right)$.

Proof.

Set $\overline{M}:=M/xM$ and assume that $\text{H}_{\mathfrak{a}}^{1}\left(\overline{M}\right)=0$. As $\mathfrak{a}$ is contained in the Jacobson radical of $R$, it follows that $\overline{M}\neq\mathfrak{a}\overline{M}$, and so $\operatorname{cd}\left(\mathfrak{a},\overline{M}\right)\geq 0$. Since $\text{H}_{\mathfrak{a}}^{1}\left(\overline{M}\right)=0$ and $\operatorname{ara}\left(\mathfrak{a},M\right)=1$, one deduces that $\operatorname{cd}\left(\mathfrak{a},\overline{M}\right)=0$. Set $T:=R/\left(\langle x\rangle+\operatorname{Ann}_{R}M\right)$. Then $\operatorname{cd}\left(\mathfrak{a},T\right)=\operatorname{cd}\left(\mathfrak{a},\overline{M}\right)=0$.

There is $y\in R$ such that

[TABLE]

So, $\text{H}_{\mathfrak{a}}^{i}\left(T\right)=\text{H}_{\langle y\rangle}^{i}\left(T\right)$ for every $i\geq 0$. We may and do choose $y$ in $\mathfrak{a}$. By [BS, Remark 2.2.20], there is the following exact sequence

[TABLE]

which implies that the natural map $\theta:T\longrightarrow T_{y}$ is surjective. In particular, there is $t\in T$ such that $\frac{t}{1_{T}}=\frac{1_{T}}{y}$, and so $y^{n}\left(yt-1_{T}\right)=0_{T}$ for some $n\in\mathbb{N}$. As $yt$ belongs to the Jacobson radical of $T$, $yt-1_{T}$ is a unite in $T$, and so it follows that $y^{n}\in\langle x\rangle+\operatorname{Ann}_{R}M$. Thus,

[TABLE]

∎

A special case of the next result has already been proved by Michael Hellus; see [He2] and [He1, Remark 1.2].

Lemma 2.6**.**

Let $\mathfrak{a}$ be a proper ideal of $R$ and $M$ a nonzero finitely generated $R$-module. Let $n\in\mathbb{N}$ be such that $\operatorname{cd}\left(\mathfrak{a},M\right)\leq n$ and $x_{1},\ldots,x_{n}\in\mathfrak{a}$. Consider the following conditions:

1. i)

$\operatorname{Rad}\left(\langle x_{1}\ldots,x_{n}\rangle+\operatorname{Ann}_{R}M\right)=\operatorname{Rad}\left(\mathfrak{a}+\operatorname{Ann}_{R}M\right)$. 2. ii)

The map $\text{H}_{\mathfrak{a}}^{n-i+1}\left(M/\langle x_{1}\ldots,x_{i-1}\rangle M\right)\stackrel{{\scriptstyle x_{i}}}{{\longrightarrow}}\text{H}_{\mathfrak{a}}^{n-i+1}\left(M/\langle x_{1}\ldots,x_{i-1}\rangle M\right)$ is surjective for all $i=1,\ldots,n$.

Then i) implies ii). Furthermore if $\mathfrak{a}$ is contained in the Jacobson radical of $R$, then i) and ii) are equivalent.

Proof.

i)$\Rightarrow$ii) It follows by Lemma 2.3.

ii)$\Rightarrow$i) We do induction on $n$. Assume that $n=1$. Since

[TABLE]

it suffices to show that

[TABLE]

Let $\mathfrak{p}\in\operatorname{V}\left(\langle x_{1}\rangle+\operatorname{Ann}_{R}M\right)$. Then $\mathfrak{p}\in\operatorname{Supp}_{R}M$. Set $T:=R/\operatorname{Ann}_{R}M$. Then

[TABLE]

So, $\text{H}_{\mathfrak{a}}^{1}\left(-\right)$ is a right exact endofunctor on the category of $T$-modules and $T$-homomorphisms. By the assumption, the map $\text{H}_{\mathfrak{a}}^{1}\left(M\right)\stackrel{{\scriptstyle x_{1}}}{{\longrightarrow}}\text{H}_{\mathfrak{a}}^{1}\left(M\right)$ is surjective. Now, we have the following display of $R$-isomorphisms:

[TABLE]

This shows that the natural map

[TABLE]

is surjective. But $x_{1}\in\mathfrak{p}$, and so the above map is zero. Thus, $\text{H}_{\mathfrak{a}}^{1}\left(M/\mathfrak{p}M\right)=0$. Since $\mathfrak{a}$ is contained in the Jacobson radical of $R$ and $\mathfrak{p}\in\operatorname{Supp}_{R}M$, it turns out that $M/\mathfrak{p}M\neq\mathfrak{a}\left(M/\mathfrak{p}M\right)$, and so $\Gamma_{\mathfrak{a}}\left(M/\mathfrak{p}M\right)\neq 0$. One has

[TABLE]

and so $\operatorname{Supp}_{R}\left(M/\mathfrak{p}M\right)=\operatorname{Supp}_{R}\left(R/\mathfrak{p}\right).$ This implies that

[TABLE]

Hence $\Gamma_{\mathfrak{a}}\left(R/\mathfrak{p}\right)\neq 0$, which implies that $\mathfrak{a}\subseteq\mathfrak{p}$, and so $\mathfrak{p}\in\operatorname{V}\left(\mathfrak{a}+\operatorname{Ann}_{R}M\right)$.

Next, assume that $n>1$ and the case $n-1$ is settled. Set $\overline{M}:=M/x_{1}M$. As

[TABLE]

applying the functor $\text{H}_{\mathfrak{a}}^{n}\left(-\right)$ on the exact sequence

[TABLE]

yields that $\text{H}_{\mathfrak{a}}^{n}\left(\overline{M}\right)$ is a quotient of $\text{H}_{\mathfrak{a}}^{n}\left(M\right)$. So, the map $\text{H}_{\mathfrak{a}}^{n}\left(\overline{M}\right)\stackrel{{\scriptstyle x_{1}}}{{\longrightarrow}}\text{H}_{\mathfrak{a}}^{n}\left(\overline{M}\right)$ is surjective. But this map is zero, and so $\text{H}_{\mathfrak{a}}^{n}\left(\overline{M}\right)=0$. Thus, $\operatorname{cd}\left(\mathfrak{a},\overline{M}\right)\leq n-1$. Since

[TABLE]

by the induction hypothesis

[TABLE]

Now by the argument given in the second paragraph of the proof of Lemma 2.4, we deduce that

[TABLE]

∎

Now, we are ready to present the main result of this paper.

Theorem 2.7**.**

Let $\mathfrak{a}$ be an ideal of $R$ which is contained in the Jacobson radical of $R$ and $M$ a nonzero finitely generated $R$-module. Assume that $c:=\operatorname{cd}\left(\mathfrak{a},M\right)=\operatorname{ara}\left(\mathfrak{a},M\right)$ and $x_{1},\ldots,x_{c}\in\mathfrak{a}$. Then the following are equivalent:

1. i)

$x_{1},\ldots,x_{c}$* is an $\mathfrak{a}$-Rs.o.p of $M$.* 2. ii)

The map $\text{H}_{\mathfrak{a}}^{c-i+1}\left(M/\langle x_{1},\ldots,x_{i-1}\rangle M\right)\stackrel{{\scriptstyle x_{i}}}{{\longrightarrow}}\text{H}_{\mathfrak{a}}^{c-i+1}\left(M/\langle x_{1},\ldots,x_{i-1}\rangle M\right)$ is surjective for all $i=1,\ldots,c$. 3. iii)

$\operatorname{cd}\left(\mathfrak{a},M/\langle x_{1},x_{2},\ldots,x_{i}\rangle M\right)=c-i$* for every $i=1,2,\ldots,c$.*

Proof.

For $c=0$, there is nothing to prove. So, in the rest of the argument, we assume that $c\geq 1$.

i)$\Leftrightarrow$ii) and i)$\Rightarrow$iii) are immediate by Lemmas 2.6 and 2.4; respectively.

iii)$\Rightarrow$i) We do induction on $c$. Suppose that $c=1$ and set $\overline{M}:=M/x_{1}M$. Then $\operatorname{ara}\left(\mathfrak{a},M\right)=1$ and $\text{H}_{\mathfrak{a}}^{1}\left(\overline{M}\right)\stackrel{{\scriptstyle\left(iii\right)}}{{=}}0$. So, Lemma 2.5 implies that

[TABLE]

Thus $x_{1}$ is an $\mathfrak{a}$-Rs.o.p of $M$. Next, suppose that $c>1$ and the claim holds for $c-1$. One has $\operatorname{cd}\left(\mathfrak{a},\overline{M}\right)\stackrel{{\scriptstyle\left(iii\right)}}{{=}}c-1$ and, for each $2\leq i\leq c$,

[TABLE]

Hence by the induction hypothesis, the sequence $x_{2},x_{3},\ldots,x_{c}$ forms an $\mathfrak{a}$-Rs.o.p of $\overline{M}$. Thus

[TABLE]

which implies that

[TABLE]

Therefore, $x_{1},\ldots,x_{c}$ is an $\mathfrak{a}$-Rs.o.p of $M$. ∎

Next, we record the following immediate conclusion which may be interesting in its own right.

Corollary 2.8**.**

Let $\left(R,\mathfrak{m}\right)$ be a local ring, $M$ a $d$-dimensional nonzero finitely generated $R$-module and $x_{1},\ldots,x_{d}\in\mathfrak{m}$. Then the following are equivalent:

1. i)

$x_{1},\ldots,x_{d}$* is a system of parameters of $M$.* 2. ii)

The map $\text{H}_{\mathfrak{m}}^{d-i+1}\left(M/\langle x_{1},\ldots,x_{i-1}\rangle M\right)\stackrel{{\scriptstyle x_{i}}}{{\longrightarrow}}\text{H}_{\mathfrak{m}}^{d-i+1}\left(M/\langle x_{1},\ldots,x_{i-1}\rangle M\right)$ is surjective for all $i=1,\ldots,d$.

Let $\left(R,\mathfrak{m}\right)$ be a local ring. Next, we will mention two results for system of parameters that their analogues don’t hold for relative system of parameters; see Example 2.9.

First: Every $R$-regular sequence is a part of a system of parameters of $R$.

Second: Let $M$ be a maximal Cohen-Maculay $R$-module and $A$ be a square matrix of size $n$ with entries in $R$. Let $x_{1},x_{2},\ldots,x_{n}$ be a system of parameters of $M$ and $y_{1},y_{2},\ldots,y_{n}\in\mathfrak{m}$ be such that $[y_{1},y_{2},\ldots,y_{n}]^{T}=A[x_{1},x_{2},\ldots,x_{n}]^{T}$. Then by [DR, Theorem], $y_{1},y_{2},\ldots,y_{n}$ is a system of parameters of $M$ if and only if the map induced by multiplication by det $A$ from $M/\langle x\rangle M$ to $M/\langle y\rangle M$ is injective.

Example 2.9**.**

Let $K$ be a field, $R=K[[x,z]]$ and $\mathfrak{a}=\langle x\rangle$. Then $\operatorname{ara}\left(\mathfrak{a}\right)=\operatorname{cd}\left(\mathfrak{a},R\right)=1$. Set $y:=zx$. Then

1. i)

One has $y\in\mathfrak{a}\setminus Z_{R}\left(R\right)$ and $x\notin\operatorname{Rad}\left(\langle y\rangle\right)$. Hence, $\text{H}_{\mathfrak{a}}^{1}\left(\frac{R}{\langle y\rangle}\right)\neq 0$ by Lemma 2.5. So, by Lemma 2.4, $y$ is not an $\mathfrak{a}$-Rs.o.p of $R$. 2. ii)

The natural map $R/\langle x\rangle\overset{z}{\longrightarrow}R/\langle y\rangle$ is injective, $x$ is an $\mathfrak{a}$-Rs.o.p of $R$ and $[y]=[z][x]$, while $y$ is not an $\mathfrak{a}$-Rs.o.p of $R$.

3. Question 1.2

Our main result in this section is Theorem 3.3. To prove it, we need the following lemma.

Lemma 3.1**.**

Let $\mathfrak{a}=\langle x_{1},\ldots,x_{n}\rangle$ be an ideal of $R$ and $M$ a finitely generated $R$-module with $\mathfrak{a}M\neq M$. Set $g:=\operatorname{grade}\left(\mathfrak{a},M\right)$. Then

1. i)

$\mathfrak{a}$* can be generated by elements $y_{1},\ldots,y_{n}$ such that $y_{i_{1}},\ldots,y_{i_{h}}$ forms an $M$-regular sequence for all $i_{1},\ldots,i_{h}$ with $1\leq i_{1}<\cdots<i_{h}\leq n,h\leq g$.* 2. ii)

If $\mathfrak{a}$ is contained in the Jacobson radical of $R$ and $g=n$, then $x_{1},\ldots,x_{n}$ forms an $M$-regular sequence.

Proof.

i) Follows by [Ka, Theorem 125 (b)].

ii) Follows by [Ka, Theorem 129]. ∎

Here is the right place to bring the following immediate corollary of Lemma 2.2.

Corollary 3.2**.**

Let $M$ be a finitely generated $R$-module and $\mathfrak{a}$ an ideal of $R$ with $M\neq\mathfrak{a}M$. Then the following are equivalent:

1. i)

$M$* is $\mathfrak{a}$-RCM and it possesses an $\mathfrak{a}$-Rs.o.p.* 2. ii)

$\operatorname{grade}\left(\mathfrak{a},M\right)=\operatorname{ara}\left(\mathfrak{a},M\right)$.

Proof.

It is clear by Lemma 2.2 and the inequality $\operatorname{grade}\left(\mathfrak{a},M\right)\leq\operatorname{cd}\left(\mathfrak{a},M\right)\leq\operatorname{ara}\left(\mathfrak{a},M\right)$. ∎

Theorem 3.3**.**

Let $M$ be a finitely generated $R$-module and $\mathfrak{a}$ an ideal of $R$ with $\operatorname{ara}\left(\mathfrak{a},M\right)=\operatorname{cd}\left(\mathfrak{a},M\right)$. Consider the following conditions:

1. i)

$M$* is $\mathfrak{a}$-RCM.* 2. ii)

Every $\mathfrak{a}$-Rs.o.p of $M$ is an $M$-regular sequence. 3. iii)

There exists an $\mathfrak{a}$-Rs.o.p of $M$ which is an $M$-regular sequence.

Then i) and iii) are equivalent. Furthermore if $\mathfrak{a}$ is contained in the Jacobson radical of $R$, all three conditions are equivalent.

Proof.

Set $c:=\operatorname{cd}\left(\mathfrak{a},M\right)$.

i)$\Rightarrow$iii) Let $y_{1},y_{2},\ldots,y_{c}\in\mathfrak{a}$ be an $\mathfrak{a}$-Rs.o.p of $M$ and set $J:=\langle y_{1},y_{2},\ldots,y_{c}\rangle$. Then

[TABLE]

and so

[TABLE]

By Lemma 3.1 i), there exist $x_{1},x_{2},\ldots,x_{c}\in R$ which forms an $M$-regular sequence and $J=\langle x_{1},x_{2},\ldots,x_{c}\rangle$. Now, $x_{1},x_{2},\ldots,x_{c}$ is our desired $\mathfrak{a}$-Rs.o.p of $M$.

iii)$\Rightarrow$i) Let $\underline{z}=z_{1},\ldots,z_{c}\in\mathfrak{a}$ be an $\mathfrak{a}$-Rs.o.p of $M$ which is an $M$-regular sequence. Then

[TABLE]

So, $M$ is $\mathfrak{a}$-RCM by Corollary 3.2.

ii)$\Rightarrow$iii) It is obvious.

i)$\Rightarrow$ii) Let $\underline{z}=z_{1},z_{2},\ldots,z_{c}\in\mathfrak{a}$ be an $\mathfrak{a}$-Rs.o.p of $R$ and set $J:=\langle z_{1},z_{2},\ldots,z_{c}\rangle$. Then

[TABLE]

and so

[TABLE]

Thus Lemma 3.1 ii) yields that $\underline{z}$ is an $M$-regular sequence. ∎

The following example shows that in Theorem 3.3, the assumption that $\mathfrak{a}$ is contained in the Jacobson radical of $R$ is necessary.

Example 3.4**.**

Let $K$ be a field. Consider the ring $R=K[x,y,z]$ and let $\mathfrak{a}=\langle x,y,z\rangle$. It is clear that $R$ is $\mathfrak{a}$-RCM. We can see that $\mathfrak{a}=\langle y\left(1-x\right),z\left(1-x\right),x\rangle$, so that $y\left(1-x\right),z\left(1-x\right),x$ is an $\mathfrak{a}$-Rs.o.p of $R$. But $y\left(1-x\right),z\left(1-x\right),x$ is not an $R$-regular sequence.

Next, we record the following corollary of Theorem 3.3.

Corollary 3.5**.**

Let $\mathfrak{a}$ be an ideal of $R$ which is contained in the Jacobson radical of $R$. Let $M$ be an $\mathfrak{a}$-RCM $R$-module and $\underline{x}=x_{1},x_{2},\ldots,x_{c}\in\mathfrak{a}$ an $\mathfrak{a}$-Rs.o.p of $M$. Then $M/\langle x_{1},x_{2},\ldots,x_{i}\rangle M$ is $\mathfrak{a}$-RCM for every $i=1,\ldots,c$.

Proof.

Set $\overline{M}:=M/\langle x_{1},x_{2},\ldots,x_{i}\rangle M$. By Theorem 3.3, $x_{1},x_{2},\ldots,x_{c}$ is an $M$-regular sequence, and so $x_{i+1},x_{i+2},\ldots,x_{c}$ is an $\overline{M}$-regular sequence. On the other hand, by Lemma 2.4, the sequence $x_{i+1},x_{i+2},\ldots,x_{c}$ is an $\mathfrak{a}$-Rs.o.p of $\overline{M}$. Applying Theorem 3.3 again implies that $\overline{M}$ is $\mathfrak{a}$-RCM. ∎

**Acknowledgement **.

The authors thank Sara Saeedi Madani for introducing them to the references [Ba1], [Ba2] and [SV].