Coartinianess of extension and torsion functors

TL;DR

This paper studies the coartinian properties of Ext and Tor modules over a noetherian local ring, establishing conditions under which these modules are coartinian based on the properties of the involved modules.

Contribution

It provides new criteria for the coartinianess of Ext and Tor modules when modules have specific compactness and dimension properties.

Findings

Ext modules are I-coartinian if M is linearly compact and N is I-cofinite with dim ≤ 1.

Tor modules are I-coartinian if M is semi-discrete linearly compact and N is finitely generated with dim ≤ 2.

Abstract

Let $(R,\mathfrak{m})$ be a commutative noetherian local ring with $\mathfrak{m}$-adic topology, $I$ an ideal of $R$. We investigate coartinianess of $\mathrm{Ext}$ and $\mathrm{Tor}$, show that the $R$-module $\mathrm{Ext}_{R}^{i}(N,M)$ is $I$-coartinian if $M$ is a linearly compact $I$-coartinian $R$-module and $N$ is an $I$-cofinite $R$-module of dimension at most $1$; the $R$-module $\mathrm{Tor}_{i}^{R}(N,M)$ is $I$-coartinian in the case $M$ is semi-discrete linearly compact $I$-coartinian and $N$ is finitely generated with dimension at most $2$.