The finiteness dimension of modules and relative Cohen-Macaulayness

TL;DR

This paper investigates the finiteness dimension and adjusted depth of modules relative to ideals in a Noetherian ring, focusing on modules that are Cohen-Macaulay relative to a specific ideal, and explores their applications.

Contribution

It introduces and analyzes the concepts of finiteness dimension and adjusted depth for modules relative to ideals, especially in the context of relative Cohen-Macaulay modules, providing new insights and applications.

Findings

Characterization of finiteness dimension for relative Cohen-Macaulay modules.

Relationships between adjusted depth and finiteness dimension.

Applications to module theory and ideal-related properties.

Abstract

Let $R$ be a commutative Noetherian ring, $\mathfrak a$ and $\mathfrak b$ ideals of $R$. In this paper, we study the finiteness dimension $f_{\mathfrak a}(M)$ of $M$ relative to $\mathfrak a$ and the $\mathfrak b$-minimum $\mathfrak a$-adjusted depth $\lambda_{\mathfrak a}^{\mathfrak b}(M)$ of $M$, where the underlying module $M$ is relative Cohen-Macaulay w.r.t $\mathfrak a$. Some applications of such modules are given.

Equations

$$f_{\mathfrak{a}}(M):=\inf \{i\in {\mathbb{N}}_0:H_{\mathfrak{a}}^i(M)\text{is not finitely generated}\},$$

$$f_{\mathfrak{a}}^{\mathfrak{b}}(M):=\inf \{i\in {\mathbb{N}}_0:\mathfrak{b}⊈\mathrm{Rad}(0{:}_R{H}_{\mathfrak{a}}^i(M))\},$$

$${\lambda }_{\mathfrak{a}}^{\mathfrak{b}}(M):=\inf \{0pt{M}_{\mathfrak{p}}+0pt(\mathfrak{a}+\mathfrak{p})/\mathfrak{p}:\mathfrak{p}\in \mathrm{Spec}(R)\setminus V(\mathfrak{b})\},$$

$${\mathfrak{a}}^{\prime }(M):={\mathfrak{a}}_0^{\prime }(M)\dots {\mathfrak{a}}_{h-1}^{\prime }(M){\mathfrak{a}}_{h+1}^{\prime }(M)\dots {\mathfrak{a}}_d^{\prime }(M),$$

$$K_{\mathfrak{a}}^i(M):=D_R(H_{\mathfrak{a}}^i(M))={\mathrm{Hom}}_R(H_{\mathfrak{a}}^i(M),E_R(k)).$$

$$0⟶M_1⟶M⟶M/M_1⟶0,$$

$$\begin{array}[]{ll}\ldots&\longrightarrow\operatorname{Ext}^{j}_{R}(M/M_{1},w_{R})\longrightarrow\operatorname{Ext}^{j}_{R}(M,w_{R})\longrightarrow\operatorname{Ext}^{j}_{R}(M_{1},w_{R})\\ &\longrightarrow\operatorname{Ext}^{j+1}_{R}(M/M_{1},w_{R})\longrightarrow\ldots\end{array}$$

$$\begin{array}[]{ll}0&\longrightarrow\operatorname{Ext}^{c-c_{1}}_{R}(M/M_{1},K(R))\longrightarrow\operatorname{Ext}^{c-c_{1}}_{R}(M,K(R))\longrightarrow\operatorname{Ext}^{c-c_{1}}_{R}(M_{1},K(R))\\ &\longrightarrow\operatorname{Ext}^{c-c_{1}+1}_{R}(M/M_{1},K(R))\longrightarrow\operatorname{Ext}^{c-c_{1}+1}_{R}(M,K(R))\longrightarrow 0,\end{array}$$

$${\mathfrak{a}}^{\prime }(M):={\mathfrak{a}}_0^{\prime }(M)\dots {\mathfrak{a}}_{h-1}^{\prime }(M){\mathfrak{a}}_{h+1}^{\prime }(M)\dots {\mathfrak{a}}_d^{\prime }(M),$$

$${\mathfrak{a}}_i^{\prime }(M)={\mathfrak{a}}_i^{\prime }(M_i)={\mathfrak{a}}_i^{\prime }({\mathcal{M}}_i)$$

$${\lambda }_{\mathfrak{a}}^{\mathfrak{b}}(M)\le 0p{t}_M(\mathfrak{a}).$$

$$H_{\mathfrak{a}}^i(M{)}_{\mathfrak{q}}\cong H_{(\mathfrak{a}+{\mathrm{Ann}}_{R}M)R_{\mathfrak{q}}}^i(M_{\mathfrak{q}})\cong H_{\mathfrak{q}{R}_{\mathfrak{q}}}^i(M_{\mathfrak{q}})$$

$$0pt{M}_{\mathfrak{q}}=\dim M_{\mathfrak{q}}.$$

$$0pt{M}_{\mathfrak{q}}=0p{t}_M(\mathfrak{a}).$$

$$f_{\mathfrak{a}}^{\mathfrak{b}}(M)={\lambda }_{\mathfrak{a}}^{\mathfrak{b}}(M).$$

$$0p{t}_M(\mathfrak{a})\le {\lambda }_{\mathfrak{a}}^{\mathfrak{b}}(M)\le 0p{t}_M(\mathfrak{a})$$

$$\mathfrak{m}-{\mathrm{fgrade}}_M(\mathfrak{a})=\min \{i\in {\mathbb{N}}_0\mid H_{\mathfrak{a}}^i(M)\text{is not Artinian}\}.$$

$$f_{\mathfrak{a}}(M)=0p{t}_M(\mathfrak{a})=\mathfrak{m}-{\mathrm{fgrade}}_M(\mathfrak{a}).$$

$$0⟶H_{Rx}^1(H_{\mathfrak{a}}^{h-1}(M))⟶H_{\mathfrak{m}}^h(M)⟶H_{Rx}^0(H_{\mathfrak{a}}^h(M))⟶0$$

Full text

1. Introduction

Throughout this paper, $R$ is a commutative Noetherian ring with identity of positive Krull dimension $n$, ${\mathfrak{a}}$ and ${\mathfrak{b}}$ two ideals of $R$ and $M$ is a finite $R$-module of positive dimension $d$. We denote by $H^{i}_{{\mathfrak{a}}}(M)$ the ith local cohomology module of $M$ supported in ${\mathfrak{a}}$. Before formulating the results of the paper we recall some related concepts. Based on [17, Definition 2.2], a finite $R$-module $M$ is called relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$ if there is precisely one non-vanishing local cohomology module of $M$ w.r.t ${\mathfrak{a}}$. Clearly this is the case if and only if $\operatorname{grade}({\mathfrak{a}},M)=\operatorname{cd}({\mathfrak{a}},M)$, where $\operatorname{cd}({\mathfrak{a}},M)$ is the largest integer $i$ for which $H^{i}_{{\mathfrak{a}}}(M)\neq 0$ and $\operatorname{grade}({\mathfrak{a}},M)$ is the least integer $i$ for which $H^{i}_{{\mathfrak{a}}}(M)\neq 0$. This concept has a connection with a notion which has been studied under the title of cohomologically complete intersection ideals in [12]. The present authors studied some properties of relative Cohen-Macaulay rings and modules in [14] and [15].

In view of [3, Definition 2.1], the increasing filtration $\mathcal{M}=\{M_{i}\}^{c}_{i=0}$ of submodules of $M$ with $c=\operatorname{cd}({\mathfrak{a}},M)$ is called the cohomological dimension filtration of $M$ if for all integer $0\leq i\leq c$, $M_{i}$ is the largest submodule of $M$ such that $\operatorname{cd}({\mathfrak{a}},M_{i})\leq i$. Recently, in [14], we have defined and studied the concept of relative Cohen-Macaulay filtered modules w.r.t ${\mathfrak{a}}$ (abbreviated as RCMF modules w.r.t ${\mathfrak{a}}$) as follows. Let $\mathcal{M}=\{M_{i}\}^{c}_{i=0}$ be the cohomological dimension filtration of submodules of $M$, where $c=\operatorname{cd}({\mathfrak{a}},M)$. Then $M$ is said to be a relative Cohen-Macaulay filtered module (relative sequentially Cohen-Macaulay module) w.r.t ${\mathfrak{a}}$, whenever $\mathcal{M}_{i}={M_{i}}/{M_{i-1}}$ is either zero or an $i$-cohomological dimensional relative Cohen-Macaulay module w.r.t ${\mathfrak{a}}$ for all $1\leq i\leq c$ (see [14, Definition 2.3]).

Recall that the finiteness dimension $f_{{\mathfrak{a}}}(M)$ of $M$ relative to ${\mathfrak{a}}$ is defined as

[TABLE]

if there exists and $+\infty$ otherwise. Another formulation is the ${\mathfrak{b}}$-finiteness dimension of $M$ relative to ${\mathfrak{a}}$ which is defined as

[TABLE]

which is either a positive integer or $+\infty$.

Moreover, the ${\mathfrak{b}}$-minimum ${\mathfrak{a}}$-adjusted depth $\lambda_{{\mathfrak{a}}}^{{\mathfrak{b}}}(M)$ of $M$ is defined as

[TABLE]

which is either a positive integer or $+\infty$ where $\operatorname{V}({\mathfrak{b}})$ denotes the set of prime ideals containing ${\mathfrak{b}}$.

The paper is divided into 3 sections. In section 2, we study vanishing and relative Cohen- Macaulayness of Ext-functors for RCMF modules with respect to an ideal. In fact, in [13, Theorem 1.4], the Peskine’s characterization of sequentially Cohen-Macaulay modules is presented in terms of Ext-groups in the graded case. Also, in [14, Definition 2.10], we have defined cohomological canonical module of $M$ as $\operatorname{Hom}_{R}(H^{\operatorname{cd}({\mathfrak{a}},M)}_{{\mathfrak{a}}}(M),E_{R}(k))$ an denoted it by $K(M)$. As a main result of this paper we prove the following result.

Let $(R,{\mathfrak{m}})$ be a relative Cohen-Macaulay local ring w.r.t ${\mathfrak{a}}$ with $\operatorname{cd}({\mathfrak{a}},R)=c$ and $K(R)$ be its cohomological canonical module. Assume that $M$ is an RCMF module w.r.t ${\mathfrak{a}}$ with the filtration in Lemma 2.4 and $c_{i}=\operatorname{cd}({\mathfrak{a}},M_{i}/M_{i-1})$ for all $1\leq i\leq r$. Then $\operatorname{Ext}^{c-c_{i}}_{R}(M,K(R))\cong\operatorname{Ext}^{c-c_{i}}_{R}(M_{i}/M_{i-1},K(R))$ is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$ of cohomological dimension $c_{i}$ for some $1\leq i\leq r$ and $\operatorname{Ext}^{j}_{R}(M,K(R))=0$ if $j\notin\{c-c_{1},\ldots,c-c_{r}\}$. Consequently, in Corollary 2.6, we partially show a similar result for RCMF $R$-modules w.r.t ${\mathfrak{a}}$, where $R$ is an arbitrary commutative Noetherian ring. More precisely, we show that if $(R,{\mathfrak{m}})$ is a local ring which is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$ with $\operatorname{cd}({\mathfrak{a}},R)=c$ and $M$ is an RCMF module w.r.t ${\mathfrak{a}}$, then the modules $\operatorname{Ext}^{c-i}_{R}(M,K(R))$ are either zero for all $0\leq i\leq\operatorname{cd}({\mathfrak{a}},M)$ or relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$ of cohomological dimension $i$.

Section 3 is devoted to examine the behavior of the notion $f_{{\mathfrak{a}}}(M)$ more closely as the annihilators of local cohomology modules is of crucial importance for researches in local cohomology. In analogy with the notation ${\mathfrak{a}}(M)={{\mathfrak{a}}}_{0}(M)\ldots{{\mathfrak{a}}}_{d-1}(M)$ in a local ring $(R,{\mathfrak{m}})$, defined in [6] and [18], where ${{\mathfrak{a}}}_{i}(M)$ is the annihilator of $H^{i}_{{\mathfrak{m}}}(M)$ for all $0\leq i\leq d-1$, we define the following notation for an $R$-module $M$ and an ideal ${\mathfrak{a}}$ of $R$ with $h:=0pt_{M}({\mathfrak{a}})$, where $R$ is not necessarily local. We set

[TABLE]

where ${{\mathfrak{a}}}^{\prime}_{i}(M)=\operatorname{Ann}_{R}(H^{i}_{{\mathfrak{a}}}(M))$ for all $0\leq i\leq d$ and $i\neq h$ and give some results concerning this notation.

One question about the annihilation of local cohomology modules is that, which ideals annihilate the local cohomology modules, and the classical theorem of this subject is Faltings’ Annihilator Theorem (see [9]). Faltings’ Annihilator Theorem states that, if $R$ is a homomorphic image of a regular ring or if $R$ has a dualizing complex, then for every choice of ideals ${\mathfrak{a}}$ and ${\mathfrak{b}}$ of $R$, $\lambda_{{\mathfrak{a}}}^{{\mathfrak{b}}}(M)=f_{{\mathfrak{a}}}^{{\mathfrak{b}}}(M)$. There are some improvement of the conditions of this theorem (see for example [5]). In Theorem 3.7, as one of our main result of this paper, we shall show that over an arbitrary Noetherian ring $R$, the Faltings’ Annihilator Theorem of local cohomology modules holds for any relative Cohen-Macaulay $R$-module $M$ w.r.t ${\mathfrak{a}}$ whenever $\operatorname{Supp}(M/{\mathfrak{a}}M)\setminus\operatorname{V}({\mathfrak{b}})$ is not empty. As a particular case of this main result we conclude that on a Noetherian local ring $(R,\mathfrak{m})$, if $M$ is a Cohen-Macaulay $R$-module, then $f_{\mathfrak{m}}^{R}(M)=\lambda_{\mathfrak{m}}^{R}(M)$. As it is mentioned in Remark 3.9, if $M$ is a relative Cohen-Macaulay $R$-module w.r.t ${\mathfrak{a}}$, $f_{{\mathfrak{a}}}(M)$ is the largest number in which $H^{f_{{\mathfrak{a}}}(M)}_{{\mathfrak{a}}}(M)$ is not Artinian. In contrast, Proposition 3.10 presents an Artinian submodule of such modules. Corollary 3.11 determines the behavior of the finiteness dimension of modules respect to non-zerodivisors, where the underlying module is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$. Some characterizations of $f_{{\mathfrak{a}}}(-)$ are provided via ring homomorphisms (Corollaries 3.12 and 3.14). Furthermore, we present two classes of modules, “multiplication” and “semidualizing” modules that satisfy $f_{{\mathfrak{a}}}(M)=f_{{\mathfrak{a}}}(R)$ (Corollaries 3.16 and  3.18). It is well-known that if $R$ is a complete Cohen-Macaulay local ring, then $w_{R}=\operatorname{Hom}_{R}(H^{n}_{{\mathfrak{m}}}(R),E(R/{\mathfrak{m}}))$ is a canonical module of $R$. We end this section by Corollary 3.19 which shows that for a relative Cohen-Macaulay local ring $R$ w.r.t ${\mathfrak{a}}$, $f_{{\mathfrak{a}}}(w_{R})=f_{{\mathfrak{a}}}(R)$ where $w_{R}$ denotes the canonical module of $R$.

For unexplained notation and terminology about local cohomology modules, we refer the reader to [5].

2. Vanishing and relative Cohen-Macaulayness of Ext-functors based on RCMF modules

This section is devoted to deal with vanishing and relative Cohen-Macaulayness of Ext-functors for RCMF modules w.r.t ${\mathfrak{a}}$. We begin by recalling the following.

Definition 2.1**.**

(see [14, Definition 2.3]) Let $M$ be a finite $R$-module and $\mathcal{M}=\{M_{i}\}^{c}_{i=0}$ be the cohomological dimension filtration of submodules of $M$, where $c=\operatorname{cd}({\mathfrak{a}},M)$. $M$ is called a relative Cohen-Macaulay filtered module (relative sequentially Cohen-Macaulay module) w.r.t ${\mathfrak{a}}$, whenever $\mathcal{M}_{i}={M_{i}}/{M_{i-1}}$ is either zero or an $i$-cohomological dimensional relative Cohen-Macaulay module w.r.t ${\mathfrak{a}}$ for all $1\leq i\leq c$. Let us abbreviate this notion by RCMF.

Definition 2.2**.**

(see [14, Definition 2.10]) Let $(R,{\mathfrak{m}},k)$ be a local ring and $M$ be a finite $R$-module with $\operatorname{cd}({\mathfrak{a}},M)=c$. For $i\neq c$, the ith cohomological deficiency module of $M$ is defined by

[TABLE]

The module $K(M):=K^{c}_{{\mathfrak{a}}}(M)$ is called the cohomological canonical module of $M$. Note that $K^{i}_{{\mathfrak{a}}}(M)=0$ for all $i<0$ or $i>c$.

Lemma 2.3**.**

(see [17, Theorem 4.3]) Let $(R,{\mathfrak{m}})$ be a relative Cohen-Macaulay local ring w.r.t ${\mathfrak{a}}$ with $\operatorname{cd}({\mathfrak{a}},R)=c$ and $K(R)$ be its canonical module. For all $0\leq t\leq c$ and all relative Cohen-Macaulay $R$-module $M$ w.r.t ${\mathfrak{a}}$ with $\operatorname{cd}({\mathfrak{a}},M)=t$, we have

• (i)

$\operatorname{Ext}^{j}_{R}(M,K(R))=0$* for all $j\neq c-t$.*

• (ii)

$\operatorname{Ext}^{c-t}_{R}(M,K(R))$* is a relative Cohen-Macaulay module w.r.t ${\mathfrak{a}}$ of cohomological dimension $t$.*

The next lemma turns out directly from Definition 2.1.

Lemma 2.4**.**

Let $M$ be an RCMF $R$-module w.r.t ${\mathfrak{a}}$ with its filtration as $0=M_{0}\subset M_{1}\subset\ldots\subset M_{r}=M$ where $r=\operatorname{cd}({\mathfrak{a}},M)$. Then for each $0\leq i\leq r$, the module $M/M_{i}$ is RCMF w.r.t ${\mathfrak{a}}$ with the filtration $0=M_{i}/M_{i}\subset M_{i+1}/M_{i}\subset\ldots\subset M_{r}/M_{i}=M/M_{i}$.

In order to prove Corollary 2.6, we prove the following.

Theorem 2.5**.**

Let $(R,{\mathfrak{m}})$ be a relative Cohen-Macaulay local ring w.r.t ${\mathfrak{a}}$ with $\operatorname{cd}({\mathfrak{a}},R)=c$ and $K(R)$ be its cohomological canonical module. Assume that $M$ is an RCMF module w.r.t ${\mathfrak{a}}$ with the filtration in Lemma 2.4 and $c_{i}=\operatorname{cd}({\mathfrak{a}},M_{i}/M_{i-1})$ for all $1\leq i\leq r$. Then $\operatorname{Ext}^{c-c_{i}}_{R}(M,K(R))\cong\operatorname{Ext}^{c-c_{i}}_{R}(M_{i}/M_{i-1},K(R))$ is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$ of cohomological dimension $c_{i}$ for some $1\leq i\leq r$ and $\operatorname{Ext}^{j}_{R}(M,K(R))=0$ if $j\notin\{c-c_{1},\ldots,c-c_{r}\}$.

Proof.

The proof is by induction on $r$. For $r=1$, $M=M_{1}/M_{0}$ is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$. Hence, by Lemma 2.3, $\operatorname{Ext}^{j}_{R}(M,K(R))=0$ for all $j\neq c-1$ and $\operatorname{Ext}^{c-1}_{R}(M,K(R))$ is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$ of cohomological dimension one. (Note that $c_{1}=\operatorname{cd}({\mathfrak{a}},M_{1}/M_{0})$ is equal to one) Now, assume that $r>1$ and, inductively, the assertion is true for $r-1$. From the short exact sequence

[TABLE]

we get the following long exact sequence

[TABLE]

By Lemma 2.3, $\operatorname{Ext}^{j}_{R}(M_{1},K(R))=0$ for all $j\neq c-c_{1}$ and $\operatorname{Ext}^{c-c_{1}}_{R}(M_{1},K(R))$ is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$ with cohomological dimension $c_{1}$. Thus we get the following exact sequence

[TABLE]

and the isomorphisms $\operatorname{Ext}^{j}_{R}(M/M_{1},K(R))\cong\operatorname{Ext}^{j}_{R}(M,K(R))$ for all $j\neq c-c_{1},c-c_{1}+1$. On the other hand, $M/M_{1}$ is RCMF module w.r.t ${\mathfrak{a}}$ by Lemma 2.4 and $M_{1}/M_{1}\subset M_{2}/M_{1}\subset\ldots\subset M_{r}/M_{1}=M/M_{1}$ is the relative Cohen-Macaulay filtration w.r.t ${\mathfrak{a}}$ of length $r-1$. Therefore by inductive hypothesis and Lemma 2.4, $\operatorname{Ext}^{j}_{R}(M/M_{1},K(R))=0$ for $j\notin\{c-c_{2},\ldots,c-c_{r}\}$. (Note that $\operatorname{cd}({\mathfrak{a}},M_{i}/M_{1}/M_{i-1}/M_{1})$ is equal to $c_{i}$ for all $2\leq i\leq r$.) This implies that $\operatorname{Ext}^{c-c_{1}}_{R}(M/M_{1},K(R))=0$ and $\operatorname{Ext}^{c-c_{1}+1}_{R}(M/M_{1},K(R))=0$. So by the last exact sequence we have $\operatorname{Ext}^{c-c_{1}}_{R}(M,w_{R})\cong\operatorname{Ext}^{c-c_{1}}_{R}(M_{1},K(R))$ is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$ of cohomological dimension $c_{1}$ and $\operatorname{Ext}^{c-c_{1}+1}_{R}(M,K(R))=0$. Thus we get the isomorphism $\operatorname{Ext}^{j}_{R}(M/M_{1},K(R))\cong\operatorname{Ext}^{j}_{R}(M,K(R))$ for all $j\neq c-c_{1}$. This implies that $\operatorname{Ext}^{j}_{R}(M,K(R))=0$ for all $j\notin\{c-c_{1},c-c_{2},\ldots,c-c_{r}\}$. Now, by this procedure, considering the filtration $M_{i}/M_{i}\subset M_{i+1}/M_{i}\subset\ldots\subset M/M_{i}$ for $M/M_{i}$ we get the assertion by induction. ∎

Corollary 2.6**.**

Let $(R,{\mathfrak{m}})$ be a relative Cohen-Macaulay local ring w.r.t ${\mathfrak{a}}$ with $\operatorname{cd}({\mathfrak{a}},R)=c$ and $K(R)$ be its cohomological canonical module. If $M$ is an RCMF $R$-module w.r.t ${\mathfrak{a}}$, then for all $0\leq i\leq\operatorname{cd}({\mathfrak{a}},M)$, the modules $\operatorname{Ext}^{c-i}_{R}(M,K(R))$ are either zero or relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$ of cohomological dimension $i$.

If we prove the converse inclusion of Corollary 2.6, we can conclude that the finite direct sum of RCMF modules is an RCMF module. There comes naturally a question whether the converse is true or not. Of course, it is easy to show that the direct sum of two relative Cohen-Macaulay modules of the same cohomological dimension is again a relative Cohen-Macaulay module.

3. The finiteness dimension of modules and relative Cohen-Macaulayness

In this section, some subjects such as the finiteness dimension $f_{{\mathfrak{a}}}(M)$ of $M$ relative to ${\mathfrak{a}}$ and the ${\mathfrak{b}}$-minimum ${\mathfrak{a}}$-adjusted depth $\lambda_{{\mathfrak{a}}}^{{\mathfrak{b}}}(M)$ of $M$ are studied. Note that we do not assume that ${\mathfrak{b}}\subseteq{\mathfrak{a}}$ in general. Furthermore, we introduce the ideal ${{\mathfrak{a}}}^{\prime}(M)$ to describe arisen problems. We also prove that Faltings’ Annihilator Theorem holds for relative Cohen-Macaulay $R$-modules. An important tool in this section is relative Cohen-Macaulayness w.r.t an ideal. We begin by defining the following notation in comparability with the notation ${{\mathfrak{a}}}(M)$ used in [6] and [18].

Definition 3.1**.**

Let $M$ be an $R$-module and ${\mathfrak{a}}$ an ideal of $R$ with $h:=0pt_{M}({\mathfrak{a}})$. We set

[TABLE]

where ${{\mathfrak{a}}}^{\prime}_{i}(M)=\operatorname{Ann}_{R}(H^{i}_{{\mathfrak{a}}}(M))$ for all $0\leq i\leq d$ and $i\neq h$.

Remark 3.2*.*

Note that if $R$ is a relative Cohen-Macaulay local ring w.r.t ${\mathfrak{a}}$, then by the above notation we have ${{\mathfrak{a}}}^{\prime}(R)=R$. Thus one deduces immediately that $\dim(R/{{\mathfrak{a}}}^{\prime}(R))=-1$. In addition, it follows that $f_{{{\mathfrak{a}}}^{\prime}(R)}(R)=\infty$.

The following corollary presents an interesting relation to the annihilators of components of the filtration of an RCMF module.

Corollary 3.3**.**

Let $M$ be an RCMF module w.r.t ${\mathfrak{a}}$ and $\mathcal{M}=\{M_{i}\}^{c}_{i=0}$ be its cohomological dimension filtration, where $c=cd({\mathfrak{a}},M)$. Then

[TABLE]

for all $0\leq i\leq c$ with $i\neq 0pt_{M}{\mathfrak{a}}$.

Proof.

Use [14, Proposition 2.12]. ∎

Corollary 3.4**.**

Let $M$ be a relative Cohen-Macaulay $R$-module w.r.t ${\mathfrak{a}}$ with $M\neq{\mathfrak{a}}M$. Then $f_{{\mathfrak{a}}}^{{\mathfrak{a}}^{\prime}(M)}(M)=\lambda_{{\mathfrak{a}}}^{{\mathfrak{a}}^{\prime}(M)}(M)$.

Proof.

As ${\mathfrak{a}}^{\prime}(M)=R$, we have $f_{{\mathfrak{a}}}^{R}(M)=\operatorname{grade}({\mathfrak{a}},M)=\lambda_{{\mathfrak{a}}}^{R}(M)$ by [5, Exercises 9.1.9 and 9.3.3] as desired. ∎

Remark 3.5*.*

For any finite $R$-module $M$ such that $0pt_{M}({\mathfrak{a}})>0$, we have the inequality $f_{{\mathfrak{a}}}(M)\leq 0pt_{M}({\mathfrak{a}})$ (see [2, Remark 2.7(i)]). It is clear that if $M$ is relative Cohen-Macaulay $R$-module w.r.t ${\mathfrak{a}}$, then $f_{{\mathfrak{a}}}(M)=0pt_{M}({\mathfrak{a}})$. But there are some examples in which $f_{{\mathfrak{a}}}(M)<0pt_{M}({\mathfrak{a}})$. Let $k$ be a field and $R=k[X,Y^{2},XY,Y^{3}]$. The height of the ideal ${\mathfrak{a}}=(X,Y^{2},XY,Y^{3})R[Z]+ZR[Z]$ of the ring of polynomials $R[Z]$ is equal to $3$ whiles the finiteness dimension $f_{{\mathfrak{a}}}(R[Z])$ is $2$ (see [5, Example 9.5.3]).

In the following result, we provide an upper bound for $\lambda_{{\mathfrak{a}}}^{{\mathfrak{b}}}(M)$.

Proposition 3.6**.**

Let $M$ be a relative Cohen-Macaulay $R$-module w.r.t ${\mathfrak{a}}$ such that $\operatorname{Supp}(M/{\mathfrak{a}}M)\setminus\operatorname{V}({\mathfrak{b}})\neq\phi$. Then

[TABLE]

Proof.

Let ${\mathfrak{p}}\in\operatorname{Supp}(M/{\mathfrak{a}}M)\setminus\operatorname{V}({\mathfrak{b}})$ and ${\mathfrak{q}}$ be a minimal prime ideal of ${\mathfrak{a}}+\operatorname{Ann}_{R}(M)$ such that ${\mathfrak{q}}\subseteq{\mathfrak{p}}$. Then

[TABLE]

for all $i\geq 0$. By [5, Theorem 6.1.4] and since $H^{0ptM_{{\mathfrak{q}}}}_{{\mathfrak{q}}R_{{\mathfrak{q}}}}(M_{{\mathfrak{q}}})$ is not zero, we conclude that

[TABLE]

But $\dim M_{{\mathfrak{q}}}=0pt_{M}({\mathfrak{a}})$, so that

[TABLE]

Now, by definition, $\lambda_{{\mathfrak{a}}}^{{\mathfrak{b}}}(M)\leq 0ptM_{{\mathfrak{q}}}=0pt_{M}({\mathfrak{a}})$ as required. ∎

Now, we are in position, to bring our main result of this section which shows that the Faltings’ Annihilator Theorem holds for relative Cohen-Macaulay modules.

Theorem 3.7**.**

Let $R$ be a Noetherian ring and $M$ be a relative Cohen-Macaulay $R$-module w.r.t ${\mathfrak{a}}$ such that $\operatorname{Supp}(M/{\mathfrak{a}}M)\setminus\operatorname{V}({\mathfrak{b}})\neq\phi$. Then

[TABLE]

Proof.

As $M$ is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$, it is easy to see that $0pt_{M}({\mathfrak{a}})=\operatorname{cd}({\mathfrak{a}},M)=f_{{\mathfrak{a}}}^{{\mathfrak{b}}}(M)$. Therefore by [5, Theorem 9.3.7] and Proposition 3.6, we get

[TABLE]

as desired. ∎

For a particular case of Theorem 3.7, we obtain the following result.

Corollary 3.8**.**

Let $(R,{\mathfrak{m}})$ be a Noetherian local ring. If $M$ is a Cohen-Macaulay $R$-module, then $f_{{\mathfrak{m}}}^{R}(M)=\lambda_{{\mathfrak{m}}}^{R}(M)$.

Proof.

It is an immediate consequence of Theorem 3.7, since $M$ is relative Cohen-Macaulay w.r.t ${\mathfrak{m}}$. ∎

Remark 3.9*.*

Let $I$ be an ideal of $R$. Recall that a sequence $a_{1},\ldots,a_{n}$ of elements of $R$ is an $I$-filter regular $M$-sequence if $a_{i}\notin{\mathfrak{p}}$ for all ${\mathfrak{p}}\in\operatorname{Ass}(M/(a_{1},\ldots,a_{i-1})M)\setminus\operatorname{V}(I)$ ($i=1,\ldots,n$) and $\operatorname{Supp}(M/(a_{1},\ldots,a_{n})M)\setminus\operatorname{V}(I)\neq\phi$. It is well-known that the length of any maximal $I$-filter regular $M$-sequence contained in ${\mathfrak{a}}$ which is denoted by $I-\operatorname{fgrade}_{M}({\mathfrak{a}})$ is the least integer $i\geq 0$ such that $\operatorname{Supp}(H^{i}_{{\mathfrak{a}}}(M))\nsubseteq\operatorname{V}(I)$, provided $\operatorname{Supp}(M/{\mathfrak{a}}M)\setminus\operatorname{V}(I)\neq\phi$ (see [1, Theorem 1.13] and P. 38 of [20]). It is easy to see that if $(R,{\mathfrak{m}})$ is a local ring and ${\mathfrak{a}}$ is an ideal of $R$ with $\operatorname{Supp}(M/{\mathfrak{a}}M)\setminus\{{\mathfrak{m}}\}\neq\phi$, then

[TABLE]

In the above situation, if $M$ is a relative Cohen-Macaulay $R$-module w.r.t ${\mathfrak{a}}$, then since $H^{f_{{\mathfrak{a}}}(M)}_{{\mathfrak{a}}}(M)\neq 0$, we get

[TABLE]

Therefore $H^{f_{{\mathfrak{a}}}(M)}_{{\mathfrak{a}}}(M)$ is not Artinian. With the notion ${\mathfrak{q}}_{{\mathfrak{a}}}(M)$ used in [11], since the top local cohomology module $H^{\dim M}_{{\mathfrak{a}}}(M)$ is Artinian, we have $q_{{\mathfrak{a}}}(M)<\dim M$. In particular, if $H^{i}_{{\mathfrak{a}}}(M)=0$ for all $i\neq 0pt_{M}({\mathfrak{a}})$, then $q_{{\mathfrak{a}}}(M)=f_{{\mathfrak{a}}}(M)$.

Although, the top local cohomology modules are almost always non-Artinian as we have seen in Remark 3.9, we provide conditions to extract an Artinian submodule of such modules as follows.

Proposition 3.10**.**

Let $(R,{\mathfrak{m}})$ be a local ring and $M$ be a relative Cohen-Macaulay $R$-module w.r.t ${\mathfrak{a}}$ with $0pt_{M}({\mathfrak{a}})=h$ and $\dim_{R}(M/{\mathfrak{a}}M)=1$. Then $H^{0}_{Rx}(H^{h}_{{\mathfrak{a}}}(M))$ is Artinian $R$-module for some $x\in{\mathfrak{m}}$.

Proof.

The assumption $\dim(M/{\mathfrak{a}}M)=1$ guaranties the existence of $x\in{\mathfrak{m}}$ such that $\dim M/({\mathfrak{a}}+Rx)M=0$. Considering the following exact sequence

[TABLE]

[19, Corollary 3.5] and the fact that $H^{h-1}_{{\mathfrak{a}}}(M)=0$ we deduce that $H^{0}_{Rx}(H^{h}_{{\mathfrak{a}}}(M))$ is Artinian over $R$. ∎

In [16, Lemma 2.7], it is proved that $f_{{\mathfrak{a}}}(M)\leq s+f_{{\mathfrak{a}}}(M/\underline{x}M)$ where $s$ is a non-negative integer and $\underline{x}=x_{1},\ldots,x_{s}$ is an $M$-regular sequence. We now provide conditions that this inequality becomes an equality.

Corollary 3.11**.**

Let $(R,{\mathfrak{m}})$ be a local ring and $M$ be a relative Cohen-Macaulay $R$-module w.r.t ${\mathfrak{a}}$ with $c:=0pt_{M}{\mathfrak{a}}$. Assume that $\underline{x}=x_{1},\ldots,x_{s}$ is a regular sequence on both $M$ and $D_{R}(H^{c}_{{\mathfrak{a}}}(M))$. Then $f_{{\mathfrak{a}}}(M)=f_{{\mathfrak{a}}}(M/\underline{x}M)+s$.

Proof.

According to definition and [14, Corollary 2.23], we have $f_{{\mathfrak{a}}}(M)=\operatorname{cd}({\mathfrak{a}},M)$ and $f_{{\mathfrak{a}}}(M/\underline{x}M)=\operatorname{cd}({\mathfrak{a}},M/\underline{x}M)$. Now, the assertion follows by [14, Theorem 2.22]. ∎

In the remainder of this section, we present a number of corollaries about the finiteness dimension $f_{{\mathfrak{a}}}(M)$ of $M$ relative to ${\mathfrak{a}}$, where $M$ is a relative Cohen-Macaulay $R$-module w.r.t ${\mathfrak{a}}$. In particular, in two corollaries below, we characterize the behavior of it under some homomorphisms.

Corollary 3.12**.**

Assume that for all integers $k\in\mathbb{N}$ there is a ring homomorphism $\phi_{k}:R\longrightarrow R$ such that the inverse system of ideals $\{\phi_{k}({\mathfrak{a}})R\}_{k\in\mathbb{N}}$ is cofinal with $\{{{\mathfrak{a}}}^{k}\}_{k\in\mathbb{N}}$. If $R$ is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$, then $f_{{\mathfrak{a}}}(R)=f_{\phi_{k}({\mathfrak{a}})R}(R)$ for all $k\in\mathbb{N}$.

Proof.

First, as the inverse system of ideals $\{\phi_{k}({\mathfrak{a}})R\}_{k\in\mathbb{N}}$ is cofinal with $\{{{\mathfrak{a}}}^{k}\}_{k\in\mathbb{N}}$, for all $i$ there exists $j$ such that ${{\mathfrak{a}}}^{j}\subseteq\phi_{i}({\mathfrak{a}})R\subseteq{\mathfrak{a}}$. Thus $\operatorname{grade}(\phi_{k}({\mathfrak{a}})R,R)=\operatorname{grade}({\mathfrak{a}},R)$ for all $k\in\mathbb{N}$. On the other hand, $\operatorname{cd}(\phi_{k}({\mathfrak{a}})R,R)=\operatorname{cd}({\mathfrak{a}},R)$ for all $k\in\mathbb{N}$ by [7, lemma 2.1]. Thus $R$ is relative Cohen-Macaulay w.r.t $\phi_{k}({\mathfrak{a}})R$ for all $k\in\mathbb{N}$. Now, Remark 3.5 completes the proof. ∎

Example 3.13**.**

For any Noetherian ring $R$ of positive characteristic, the preceding corollary is true if we take $\phi_{k}:R\longrightarrow R$ as the Frobenius map.

Corollary 3.14**.**

Let $f:R\longrightarrow R^{\prime}$ be a faithfully flat homomorphism of Noetherian rings. If $R$ is relative Cohen-Macaulay w.r.t ${\mathfrak{a}}$, then $f_{{\mathfrak{a}}}(R)=f_{{\mathfrak{a}}R^{\prime}}(R^{\prime})$.

Proof.

By [14, Proposition 3.7], $R^{\prime}$ is a relative Cohen-Macaulay ring w.r.t ${\mathfrak{a}}R^{\prime}$. Now, apply Remark 3.5 to end the proof. ∎

To see examples for Corollary 3.14, we refer the reader to [14, Examples 3.9 and 3.10].

Now, we remind two classes of $R$-modules for which $f_{{\mathfrak{a}}}(M)=f_{{\mathfrak{a}}}(R)$.

Definition 3.15**.**

(see [8]) An $R$-module $M$ is called multiplication if for every submodule $N$ of $M$ there exists an ideal ${\mathfrak{a}}$ of $R$ such that $N={\mathfrak{a}}M$.

Corollary 3.16**.**

Let $R$ be a relative Cohen-Macaulay ring w.r.t ${\mathfrak{a}}$ and $M$ be a faithful multiplication $R$-module. Then $f_{{\mathfrak{a}}}(M)=f_{{\mathfrak{a}}}(R)$.

Proof.

Apply [14, Corollary 3.2] and Remark 3.5. ∎

Definition 3.17**.**

(see [10] and [21]) An $R$-module $M$ is semidualizing if it satisfies the following:

• (i)

$M$ is finitely generated,

• (ii)

The homotopy map ${\chi}^{R}_{M}:R\longrightarrow\operatorname{Hom}_{R}(M,M)$, defined by $r\mapsto[s\mapsto rs]$, is an isomorphism, and

• (iii)

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

Corollary 3.18**.**

Let $R$ be a relative Cohen-Macaulay ring w.r.t ${\mathfrak{a}}$ and $M$ be a semidualizing $R$-module. Then $f_{{\mathfrak{a}}}(M)=f_{{\mathfrak{a}}}(R)$.

Proof.

Apply [14, Corollary 3.4] and Remark 3.5. ∎

At the end, we obtain a result on canonical modules.

Corollary 3.19**.**

Let $(R,{\mathfrak{m}})$ be a relative Cohen-Macaulay local ring w.r.t ${\mathfrak{a}}$ and $w_{R}$ be its canonical module. Then $f_{{\mathfrak{a}}}(w_{R})=f_{{\mathfrak{a}}}(R)$.

Proof.

By [14, Proposition 3.5], the canonical module $w_{R}$ is relative Cohen-Macaulay module w.r.t ${\mathfrak{a}}$. Thus the proof is completed by Remark 3.5. ∎

Acknowledgements. The authors would like to express their thanks to Dr. Raheleh Jafari from Kharazmi University for her useful comments.