Associated primes of local cohomology modules of weakly Laskerian modules

TL;DR

This paper investigates the associated primes of local cohomology modules of weakly Laskerian modules over Noetherian rings, establishing finiteness results under various conditions, especially for principal ideals and low-dimensional rings.

Contribution

It proves finiteness of associated primes of local cohomology modules for weakly Laskerian modules in new cases, including principal ideals and low-dimensional rings.

Findings

Finiteness of associated primes when the ideal is principal.

Finiteness holds for local rings with dimension ≤ 3.

Finiteness holds when the quotient by the ideal has dimension ≤ 1.

Abstract

The notion of weakly Laskerian modules was introduced recently by the authors. Let $R$ be a commutative Noetherian ring with identity, $\fa$ an ideal of $R$, and $M$ a weakly Laskerian module. It is shown that if $\fa$ is principal, then the set of associated primes of the local cohomology module $H_{\fa}^i(M)$ is finite for all $i\geq 0$. We also prove that when $R$ is local, then $\Ass_R(H_{\fa}^i(M))$ is finite for all $i\geq 0$ in the following cases: (1) $\dim R\leq 3$, (2) $\dim R/\fa\leq 1$, (3) $M$ is Cohen-Macaulay and for any ideal $\fb$, with $l=\grade(\fb,M)$, $\Hom_R(R/\fb,H_{\fb}^{l+1}(M))$ is weakly Laskerian.