Monomial ideals whose depth function has any given number of strict local maxima

Abstract

We construct monomial ideals with the property that their depth function has any given number of strict local maxima.1991 Mathematics Subject Classification. 13A15, 13C13.

Polarization of Koszul cycles with applications to powers of edge ideals of whisker graphs

“…While it is known by a classical result of Brodmann [2] that f (k) for k ≫ 0 is constant, the behavior of f (k) is not so well understood for initial values of k. In [8], it is shown that any non-decreasing bounded integer function f (k) is the depth function of a suitable monomial ideal and it is conjectured that f (k) can be any convergent nonnegative integer valued function. In support of this conjecture, it was shown in [1] that f (k) may have arbitrarily many local maxima. On the other hand, it seems that the depth function for the edge ideals behave more tamely.…”

Depth and regularity modulo a principal ideal

“…In general, we may have Ass I t ⊆ Ass I t+1 (see e.g. [1,13,15]). It has been of great interest to know when Ass I t ⊆ Ass I t+1 for all t ≥ 1 (see e.g.…”

The behavior of depth functions of cover ideals of unimodular hypergraphs

“…Let R=k[x 1 , ..., x n ] be a polynomial ring over a given field k, and let I be a homogeneous ideal in R. It is known by Brodmann [3] that depth(R/I s ) takes a constant value for large s. Moreover, Two natural questions arise from Brodmann's theorem: (1) What is the nature of the function s →depth R/I s for s dstab(I)? (2) What is a reasonable bound for dstab(I)?…”

“…[1]). Squarefree monomial ideals behave considerably better than monomial ideals in general, and Herzog and Hibi [10] asked whether depth functions of any squarefree monomial I is non-increasing, that is depth R/I s depth R/I s+1 for all s 1.…”