Finiteness properties of formal local cohomology modules and Cohen-Macaulayness

TL;DR

This paper studies the structure and finiteness properties of formal local cohomology modules in local rings, revealing conditions under which these modules exhibit Artinian behavior and applying these results to characterize Cohen-Macaulay modules.

Contribution

It introduces new finiteness results for formal local cohomology modules and establishes criteria for Cohen-Macaulayness based on these modules.

Findings

Formal local cohomology modules are Artinian under certain conditions.

A criterion for Cohen-Macaulayness using formal local cohomology modules.

An upper bound for cohomological dimension related to formal grade.

Abstract

Let $\fa$ be an ideal of a local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. We investigate the structure of the formal local cohomology modules ${\vpl}_nH^i_{\fm}(M/\fa^n M)$, $i\geq 0$. We prove several results concerning finiteness properties of formal local cohomology modules which indicate that these modules behave very similar to local cohomology modules. Among other things, we prove that if $\dim R\leq 2$ or either $\fa$ is principal or $\dim R/\fa\leq 1$, then $\Tor_j^R(R/\fa,{\vpl}_nH^i_{\fm}(M/\fa^n M))$ is Artinian for all $i$ and $j$. Also, we examine the notion $\fgrade(\fa,M)$, the formal grade of $M$ with respect to $\fa$ (i.e. the least integer $i$ such that ${\vpl}_nH^i_{\fm}(M/\fa^n M) \neq 0$). As applications, we establish a criterion for Cohen-Macaulayness of $M$, and also we provide an upper bound for cohomological dimension of $M$ with respect to $\fa$.