Asymptotic stability of depths of localizations of modules

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 shows that for large integers 𝑛 > 0, the depth of the localizations of (𝑀/𝐼^𝑛 𝑀)_𝔭 are stable uniformly for all prime ideals 𝔭 of 𝑅 in each of the following cases: (1) 𝑅 is CM-excellent, (2) 𝑅 is semi-local, (3) 𝑀 or 𝑀/𝐼^𝑛 𝑀 for some 𝑛 > 0 is Cohen–Macaulay.