Cofiniteness of modules and local cohomology

TL;DR

This paper investigates the properties of a specific class of modules over noetherian rings and establishes conditions for their cofiniteness, especially focusing on local cohomology modules in higher-dimensional cases.

Contribution

It introduces new criteria for modules to belong to the class _n() and analyzes cofiniteness of local cohomology modules when the quotient ring has dimension at least three.

Findings

Established sufficient conditions for modules to be in _n().

Proved cofiniteness of local cohomology modules in dimension    3.

Extended understanding of module cofiniteness in higher-dimensional rings.

Abstract

Let $A$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $A$ and let $n$ be a non-negative integer. In this paper, we study $\mathcal{S}_{n}(\mathfrak{a})$, a certain class of $A$-modules and we find some sufficient conditions so that a module belongs to $\mathcal{S}_{n}(\mathfrak{a})$. Moreover, we study the cofiniteness of local cohomology modules when $\dim A/\frak a\geq 3$.