Finiteness dimensions and cofiniteness of generalized local cohomology modules

TL;DR

This paper investigates the finiteness and cofiniteness properties of generalized local cohomology modules over Noetherian rings, establishing conditions under which these modules are finite or cofinite and analyzing their associated primes.

Contribution

It introduces new criteria for the cofiniteness of generalized local cohomology modules and studies their associated primes in relation to finiteness dimensions.

Findings

Cofiniteness of local cohomology modules under certain Ext finiteness conditions.

Finiteness of associated primes of specific local cohomology modules.

Conditions linking Ext modules' finiteness to local cohomology properties.

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ and ideal of $R$, $M$ a finite $R$--module, and $n$ a non-negative integer. In this paper, for an arbitrary $R$--module $X$ which is not necessarily finite, we study the finiteness dimension $f_\mathfrak{a}(M,X)$ and the $n$-th finiteness dimension $f^n_\mathfrak{a}(M,X)$ of $M$ and $X$ with respect to $\mathfrak{a}$. Assume that $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is finite for all $i\leq f^2_\mathfrak{a}(M,X)$ (resp. $i< f^1_\mathfrak{a}(M,X)$). We show that $\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ is $\mathfrak{a}$--cofinite for all $i< f^2_\mathfrak{a}(M,X)$ (resp. $i< f^1_\mathfrak{a}(M,X)$) and $\operatorname{Ass}_{R}(\operatorname{H}^{f^2_\mathfrak{a}(M,X)}_{\mathfrak{a}}(M,X))$ (resp. if $\operatorname{Ext}^{f^1_\mathfrak{a}(M,X)}_{R}(R/\mathfrak{a},X)$ is finite, then $\operatorname{Ass}_{R}(\operatorname{H}^{f^1_\mathfrak{a}(M,X)}_{\mathfrak{a}}(M,X))$) is finite.