When does depth stabilize early on?

TL;DR

This paper investigates conditions under which the depth of powers of an ideal in a polynomial ring remains constant, providing new criteria related to Cohen-Macaulay Rees algebras and direct summands.

Contribution

It establishes sufficient conditions for the depth function to be constant, extending previous results to a broader class of ideals including square-free monomial ideals.

Findings

Depth remains constant under Cohen-Macaulay Rees algebra

Conditions involving cohomological and projective dimensions are sufficient

Includes and generalizes recent results on square-free monomial ideals

Abstract

In this paper we study graded ideals I in a polynomial ring S such that the numerical function f(k)=depth(S/I^k) is constant. We show that, if (i) the Rees algebra of I is Cohen-Macaulay, (ii) the cohomological dimension of I is not larger than the projective dimension of S/I and (iii) the K-algebra generated by some generators of I is a direct summand of S, then f(k) is constant. When I is a square-free monomial ideal, the above criterion includes as special cases all the results of a recent paper by Herzog and Vladoiu. In this combinatorial setting there is a chance that the converse of the above fact holds true.