Cominimaxness with respect to ideals of dimension one

TL;DR

This paper investigates conditions under which Ext modules are minimax over Noetherian rings, especially focusing on modules with support in ideals of dimension one, and explores implications for local cohomology and finiteness properties.

Contribution

It establishes new criteria for minimaxness of Ext modules related to ideals of dimension one, generalizing previous results and providing equivalent conditions for local cohomology modules.

Findings

Ext modules are minimax under certain conditions involving dimension and support.

Finiteness of Bass and Betti numbers for a-torsion modules when Ext modules are minimax.

Equivalent conditions for cominimaxness of local cohomology modules of dimension at most one.

Abstract

Let $R$ be a commutative Noetherian ring, $\fa$ be an ideal of $R$ and $M$ be an $R$-module. It is shown that if $\Ext^i_R(R/\fa,M)$ is minimax for all $i\leq \dim M$, then the $R$-module $\Ext^i_R(N,M)$ is minimax for all $i\geq 0$ and for any finitely generated $R$-module $N$ with $\Supp_R(N) \subseteq V (\fa)$ and $\dim N \leq 1$. As a consequence of this result we obtain that for any $\fa$-torsion $R$-module $M$ that $\Ext^i_R(R/\fa, M)$ is minimax for all $i\leq \dim M$, all Bass numbers and all Betti numbers of $M$ are finite. This generalizes \cite[Corollary 2.7]{BNS2015}. Also, some equivalent conditions for the cominimaxness of local cohomology modules with respect to ideals of dimension at most one are given.