Annihilator of local cohomology modules under localization and completion

TL;DR

This paper investigates how the annihilators of local cohomology modules relate to the structure of Noetherian local rings, characterizing properties like catenarity, unmixedness, and Cohen-Macaulay quotients through localization and completion.

Contribution

It provides new characterizations of catenarity, unmixedness, and Cohen-Macaulay quotients based on the behavior of annihilators of local cohomology modules under localization and completion.

Findings

Catenarity and unmixedness characterized via annihilators of top local cohomology

Necessary and sufficient conditions for Cohen-Macaulay quotients established

Compatibility of annihilators under localization and completion analyzed

Abstract

Let $(R, \frak m)$ be a Noetherian local ring. This paper deals with the annihilator of Artinian local cohomology modules $H^i_{\frak m}(M)$ in the relation with the structure of the base ring $R$, for non negative integers $i$ and finitely generated $R$-modules $M$. Firstly, the catenarity and the unmixedness of local rings are characterized via the compatibility of annihilator of top local cohomology modules under localization and completion, respectively. Secondly, some necessary and sufficient conditions for a local ring being a quotient of a Cohen-Macaulay local ring are given in term of the annihilator of all local cohomology modules under localization and completion.