Cofiniteness of weakly Laskerian local cohomology modules

TL;DR

This paper investigates the cofiniteness properties of certain local cohomology modules over Noetherian rings, establishing finiteness of associated primes and the Abelian category structure of specific module classes.

Contribution

It introduces the class of ${ m FD_{ ext{leq} n}}$ modules and proves cofiniteness and finiteness results for local cohomology modules under weakly Laskerian conditions.

Findings

Finiteness of associated primes of certain local cohomology modules.

Finiteness of ${ m Hom}$ and ${ m Ext}^1$ modules involving local cohomology.

The category of $I$-cofinite ${ m FD_{ ext{leq}1}}$ modules is Abelian.

Abstract

Let $I$ be an ideal of a Noetherian ring R and M be a finitely generated R-module. We introduce the class of extension modules of finitely generated modules by the class of all modules $T$ with $\dim T\leq n$ and we show it by ${\rm FD_{\leq n}}$ where $n\geq -1$ is an integer. We prove that for any ${\rm FD_{\leq 0}}$(or minimax) submodule N of $H^t_I(M)$ the R-modules ${\rm Hom}_R(R/I,H^{t}_I(M)/N) {\rm and} {\rm Ext}^1_R(R/I,H^{t}_I(M)/N)$ are finitely generated, whenever the modules $H^0_I(M)$, $H^1_I(M)$, ..., $H^{t-1}_I(M)$ are ${\rm FD_{\leq 1}}$ (or weakly Laskerian). As a consequence, it follows that the associated primes of $H^{t}_I(M)/N$ are finite. This generalizes the main results of Bahmanpour and Naghipour, Brodmann and Lashgari, Khashyarmanesh and Salarian, and Hong Quy. We also show that the category $\mathscr {FD}^1(R,I)_{cof}$ of $I$-cofinite ${\rm FD_{\leq1}}$ ~ $R$-modules forms an Abelian subcategory of the category of all $R$-modules.