Structure of modules stably annihilated by a fixed ideal

TL;DR

This paper studies modules over a noetherian ring that are annihilated by a fixed ideal, revealing their structure and extending known theorems from Gorenstein to Cohen--Macaulay rings.

Contribution

It introduces a new subcategory of modules with stable annihilators containing a given ideal and extends a theorem on syzygy categories to Cohen--Macaulay rings.

Findings

Characterization of modules with stable annihilators containing a fixed ideal

Extension of Dey and Liu's theorem to Cohen--Macaulay rings

Insights into the structure of syzygy categories in this context

Abstract

Let $R$ be a commutative noetherian ring, and denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules. In this paper, for an ideal $I$ of $R$, we introduce the full subcategory $\operatorname{mod}_{I}(R)$ of $\operatorname{mod} R$ consisting of modules whose stable annihilators contain $I$, and we investigate its structure. As an application, we explore the syzygy category of maximal Cohen--Macaulay $R$-modules, extending a theorem of Dey and Liu from the Gorenstein case to the Cohen--Macaulay case.