The eventual stability of depth, associated primes and cohomology of a graded module

TL;DR

This paper investigates the long-term stability of homological invariants like depth, associated primes, and cohomology in graded modules, providing new stability results and bounds on the stabilization degree.

Contribution

It offers new stability theorems and estimates for the degree of stabilization of homological invariants in graded modules, extending prior understanding.

Findings

Stability results for depth, associated primes, and cohomology of graded modules.

Bounds on the degree from which invariants stabilize.

Enhanced understanding of Castelnuovo-Mumford regularity in this context.

Abstract

The asymptotic stability of several homological invariants of the graded pieces of a graded module has attracted quite a lot of attention over the last decades. We provide in this text several stability results together with estimates of the degree from which it stabilizes. Before we establish these regularity results, we prove several facts about depth and cohomological dimension with respect to a finitely generated ideal and about Castelnuovo-Mumford regularity of a graded module.