When does depth stabilize early on?

Abstract

In this paper we study graded ideals I in a polynomial ring S such that the numerical function 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 homogeneous generators of I is a direct summand of S, then depth(S/I k ) is constant. All the ideals with constant depth-function discovered by Herzog and Vladoiu in [HV] satisfy the criterion given above. In the contest of square-free monomial ideals, there is a chance that a converse of the previous fact holds true.

Powers of sums and their homological invariants

“…In another direction, by results due to Cutkosky, Herzog, Kodiyalam, N.V. Trung and Wang [16,38,51], the Castelnuovo-Mumford regularity reg I s is a linear function of s for s ≫ 0. By definition, for a finitely generated graded R-module M , its Castelnuovo-Mumford regularity is reg M = sup{i + j : H i m (M ) j = 0} where H i m (M ) denotes the ith local cohomology of M with support in m. For more information on asymptotic properties of powers of ideals, see, e.g., [6,10,11,12,20,21,27,39,47] and their references.…”