Filter regular sequences and generalized local cohomology modules

TL;DR

This paper introduces the concept of filter grade for ideals on modules, characterizes properties of generalized local cohomology modules, and explores conditions for their Artinian property and attached primes.

Contribution

It defines filter grade, provides characterizations, and applies these to analyze generalized local cohomology modules, including their Artinian nature and attached primes.

Findings

Determined the least degree where local cohomology is not Artinian.

Proved local cohomology modules are Artinian iff a certain dimension condition holds.

Established the Nagel-Schenzel formula for generalized local cohomology.

Abstract

Let $\frak a$, $\frak b$ be ideals of a commutative Noetherian ring $R$ and let $M$, $N$ be finite $R$-modules. The concept of an $\frak a$-filter grade of $\frak b$ on $M$ is introduced and several characterizations and properties of this notion are given. Then, using the above characterizations, we obtain some results on generalized local cohomology modules $H^i_{\frak a}(M, N)$. In particular, first we determine the least integer $i$ for which $H^i_{\frak a}(M, N)$ is not Artinian. Then we prove that $H^i_{\frak a}(M, N)$ is Artinian for all $i\in\mathbb N_0$ if and only if $\dim{R}/({\frak a+Ann M+Ann N})=0$. Also, we establish the Nagel-Schenzel formula for generalized local cohomology modules. Finally, in a certain case, the set of attached primes of $H^i_{\frak a}(M, N)$ is determined and a comparison between this set and the set of attached primes of $H^i_{\frak a}(N)$ is given.