Asymptotic stability of depths of localizations of modules

TL;DR

This paper proves that the depth of localizations of modules M/I^n M stabilizes for large n under certain conditions on the ring R or the modules, extending understanding of asymptotic properties in commutative algebra.

Contribution

It establishes conditions under which the depth of localizations of M/I^n M stabilizes for large n, generalizing previous results in the asymptotic behavior of modules.

Findings

Depth stabilizes for large n when M or M/I^n M is Cohen-Macaulay for some n.

Depth stabilization occurs if R is a homomorphic image of a Cohen-Macaulay ring.

Results apply to rings that are semi-local, excellent, quasi-excellent, catenary, or acceptable.

Abstract

Let R be a commutative noetherian ring, I an ideal of R, and M a finitely generated R-module. The asymptotic behavior of the quotient modules M/I^n M of M is an actively studied subject in commutative algebra. The main result of this paper asserts that the depth of the localization of M/I^n M at any prime ideal of R is stable for large integers n that do not depend on the prime ideal, if the module M or M/I^n M is Cohen-Macaulay for some n>0, or the ring R is one of the following: a homomorphic image of a Cohen-Macaulay ring, a semi-local ring, an excellent ring, a quasi-excellent and catenary ring, and an acceptable ring.