Cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules

TL;DR

This paper investigates the conditions under which generalized local cohomology modules have finite associated primes and are cofinite, providing new results on their structure and finiteness properties in Noetherian rings.

Contribution

It establishes new criteria for cofiniteness and finiteness of associated primes of generalized local cohomology modules, extending previous results to broader contexts.

Findings

H^{i}_{\u03a0}(M,X) is (FD_{<n},a)-cofinite under certain conditions.

The set of associated primes with large dimension is finite for all i.

H^{i}_{a}(M,N) has finite associated primes when R is semi-local and modules have dimension 3 or less.

Abstract

Let $n$ be a non-negative integer, $R$ a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$, $M$ and $N$ two finitely generated $R$-modules, and $X$ an arbitrary $R$-module. In this paper, we study cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules. In some cases, we show that $\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ is an $(\operatorname{FD}_{<n},\mathfrak{a})$-cofinite $R$-module and $\{\mathfrak{p}\in\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M,X)):\dim(R/\mathfrak{p})\geq{n}\}$ is a finite set for all $i$. If $R$ is semi-local, we observe that $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M,N))$ is finite for all $i$ when $\dim_R(M)\leq{3}$ or $\dim_R(N)\leq{3}$. Also, in some situations, we prove that $\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ is an $\mathfrak{a}$-cofinite $R$-module for all $i$.