Narrow Results

Let $𝔞$ be an ideal of a commutative noetherian ring $R$ and $M,N$ two $R$-modules with $M$ finitely generated. It is shown that if either ${\mathrm{\text{H}}}_{𝔞}^i(N)$ is an $𝔞$-cofinite module of dimension $\le 1$ for all $i$, or $𝔞$ is a principal ideal and ${\mathrm{\text{Ext}}}_R^i(R/𝔞,N)$ is finitely generated for all $i$, or ${\mathrm{\text{Ext}}}_R^i(R/𝔞,N)$ is finitely generated and ${\mathrm{\text{dim}}}_R{\mathrm{\text{H}}}_{𝔞}^i(M,N)\le 1$ for all $i$, then the $R$-module ${\mathrm{\text{H}}}_{𝔞}^t(M,N)$ is $𝔞$-cofinite for all $t\ge 0$.

Let $(R,𝔪)$ be a commutative Noetherian local ring, which is a homomorphic image of a Gorenstein local ring and $I$ an ideal of $R$. Let $M$ be a nonzero finitely generated $R$-module and $i\ge 0$ be an integer. In this paper we show that, the $R$-module $H_{𝔪}^i(M)$ is nonzero and $I$-cofinite if and only if $Rad(I+0{:}_R{H}_{𝔪}^i(M))=𝔪$. Also, several applications of this result will be included.

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $𝔞$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $𝔞$, if and only if, ${Ext}_R^i(R/𝔞,M)$ is finitely generated for all $i\le \mathrm{\text{cd}}(𝔞,M)+1$, whenever $dimR/𝔞=1$. In addition, we show that if $M$ is finitely generated and $H_{𝔞}^i(M)$ are weakly Laskerian for all $i\le t-1$, then $H_{𝔞}^i(M)$ are $𝔞$-cofinite for all $i\le t-1$ and for any minimax submodule $K$ of $H_{𝔞}^t(M)$, the $R$-modules ${Hom}_R(R/𝔞,H_{𝔞}^t(M)/K)$ and ${Ext}_R^1(R/𝔞,H_{𝔞}^t(M)/K)$ are finitely generated, where $t$ is a non-negative integer. Finally, we explore a criterion for weakly cofiniteness of modules with respect to an ideal of dimension one. Namely, for such ideals it suffices that the two first $Ext$-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result, we deduce that the category of all $𝔞$-weakly cofinite modules over $R$ forms a full Abelian subcategory of the category of modules.

Let $R$ be a commutative noetherian ring, and ${𝒵}$ a stable under specialization subset of $Spec(R)$. We introduce a notion of ${𝒵}$-cofiniteness and study its main properties. In the case $dim({𝒵})\le 1$, or $dim(R)\le 2$, or $R$ is semilocal with $cd({𝒵},R)\le 1$, we show that the category of ${𝒵}$-cofinite $R$-modules is abelian. Also, in each of these cases, we prove that the local cohomology module $H_{{𝒵}}^i(X)$ is ${𝒵}$-cofinite for every homologically left-bounded $R$-complex $X$ whose homology modules are finitely generated and every $i\in \mathbb{Z}$.

Let $R$ be a commutative Noetherian ring and let $I$ and $J$ be ideals of $R$. The main purpose of this paper is to compare of the finiteness properties of $H_I^i(R)$ and $H_I^i(R\bowtie J)$, where $H_I^i$ is the $i$th local cohomology module functor with respect to $I$ and $R\bowtie J$ is the amalgamated of $R$ along $J$.

Let $(R,𝔪)$ be a commutative Noetherian complete local ring and $I$ be a proper ideal of $R$. Suppose that $M$ is a nonzero $I$-cofinite $R$-module of Krull dimension $n$. In this paper, it shown that $dimR/(I+{Ann}_R{H}_{𝔪}^n(M))=n.$ As an application of this result, it is shown that $dimR/(I+𝔭)=n$, for each $𝔭\in {Att}_R{H}_{𝔪}^n(M).$ Also it shown that for each $j\ge 0$ the submodule ${\mathrm{\Sigma }}_j(M):=\cup \{K:K\le M$ and $dimK\le j\}$ of $M$ is $I$-cofinite, $dimR/(I+{Ann}_R{H}_{𝔪}^j(M/{\mathrm{\Sigma }}_j(M)))<j$ and $dimR/(I+{Ann}_R{H}_{𝔪}^j(M))\le j,$ whenever the category of all $I$-cofinite $R$-modules is an Abelian subcategory of the category of all $R$-modules. Also some applications of these results will be included.

Let ${𝒵}$ be a specialization closed subset of $\mathrm{\text{Spec}}R$ and ${𝒮}$ be a Serre subcategory of $ModR$. As a generalization of the notion of cofiniteness, we introduce the concept of ${𝒵}$-coserreness with respect to ${𝒮}$ (see Definition 4.1). First, as a main result, for some special Serre subcategories ${𝒮}$, we show that an $R$-module $M$ with ${dim}_{R}M\le 1$ is ${𝒵}$-coserre with respect to ${𝒮}$ if and only if ${Ext}_R^{0,1}(R/𝔞,M)\in {𝒮}$ for all ideals $𝔞\in F({𝒵})$. Indeed, this result provides a partial answer to a question that was recently raised in [K. Bahmanpour, R. Naghipour and M. Sedghi, Modules cofinite and weakly cofinite with respect to an ideal, J. Algebra Appl. 16 (2018) 1–17]. As an application of this result, we show that the category of ${𝒵}$-coserre $R$-modules $M$ with ${dim}_{R}M\le 1$ is a full Abelian subcategory of $ModR$. Also, for every homologically bounded $R$-complex $X$ whose homology modules belong to ${𝒮}$ we show that the local cohomology modules ${\mathrm{\text{H}}}_{{𝒵}}^i(X)$ for all $i$, are ${𝒵}$-coserre in all the cases where ${dim}_R{𝒵}\le 1$, ${dim}_{R}X\le 2-supX$ and $cd({𝒵},X)\le 1-supX$.

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$ such that the $R$-modules $H_I^i(M)$ are $I$-cofinite, for all finitely generated $R$-modules $M$ and all $i\in {\mathbb{N}}_0$. In this paper, we investigate a question of Hartshorne concerning the Abelianness of the category of all $I$-cofinite modules.

We show that for a linearly compact module over a local ring (R,𝔪), the concept of co-support of modules provided by Melkersson is equivalent with Yassemi's concept. We also introduce the definition of coartinian modules which is in some sense dual to Hartshorne's definition of cofinite modules and show that is I-coartinian in case M is a semi-discrete linearly compact R-module with Ndim M=d> 0.

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M an R-module. We show that, whenever , M is Noetherian if and only if there exists a submodule N of M such that the R-modules M/𝔞 N and are Noetherian. By using this result, we establish Noetherian properties for local cohomology modules in several cases. For instance, we obtain a new version of the Lichtenbaum-Hartshorne Vanishing Theorem.