Depth functions of symbolic powers of homogeneous ideals

Abstract

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function depthR/I(t)=dimR-pdI(t)-1, where I(t) denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and pd denotes the projective dimension. It has been an open question whether the function depthR/I(t) is non-increasing if I is a squarefree monomial ideal. We show that depthR/I(t) is almost non-increasing in the sense that depthR/I(s)≥depthR/I(t) for all s≥ 1 and t∈ E(s) , where E(s)=⋃i≥1{t∈N|i(s-1)+1≤t≤is}(which contains all integers t≥ (s- 1) 2+ 1). The range E(s) is the best possible since we can find squarefree monomial ideals I such that depthR/I(s)