Asymptotic stability of depths of localizations of modules

Abstract

Let $$ be a commutative Noetherian ring, $$ an ideal of $$, and $$ a finitely generated $$-module. The asymptotic behavior of the quotient modules $$ of $$ is an actively studied subject in commutative algebra. The main result of this paper shows that for large integers $$, the depth of the localizations of $$ are stable uniformly for all prime ideals $$ of $$ in each of the following cases: (1) $$ is CM-excellent, (2) $$ is semi-local, (3) $$ or $$ for some $$ is Cohen–Macaulay.