On the cofiniteness of generalized local cohomology modules

TL;DR

This paper investigates the conditions under which generalized local cohomology modules are cofinite, proving cofiniteness for principal ideals, low-dimensional supports, and modules of small dimension.

Contribution

It establishes new criteria for the cofiniteness of generalized local cohomology modules in various algebraic settings.

Findings

Cofiniteness holds for principal ideals for all modules and degrees.

Modules with support dimension at most 1 have cofiniteness in certain degrees.

Modules of dimension at most 2 have cofiniteness in all degrees.

Abstract

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$, $N$ two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $\displaystyle H^j_I(M,N)=\dlim\Ext^j_R(M/I^nM,N)$ of $M$ and $N$ with respect to $I$. We first prove that if $I$ is a principal ideal then $H^j_I(M,N)$ is $I$-cofinite for all $M, N$ and all $j$. Secondly, let $t$ be a non-negative integer such that $\dim\Supp(H^j_I(M,N))\le 1 \text{for all} j<t.$ Then $H^j_I(M,N)$ is $I$-cofinite for all $j<t$ and $\Hom(R/I,H^t_I(M,N))$ is finitely generated. Finally, we show that if $\dim(M)\le 2$ or $\dim(N)\le 2$ then $H^j_I(M,N)$ is $I$-cofinite for all $j$.