Remarks on the Stanley depth and Hilbert depth of monomial ideals with linear quotients

TL;DR

This paper investigates the Stanley and Hilbert depths of monomial ideals with linear quotients, establishing conditions under which these depths equal the module's depth and providing inequalities relating them.

Contribution

It proves that for monomial ideals with linear quotients and certain depth conditions, the Stanley and Hilbert depths match the module's depth, extending understanding of their relationship.

Findings

sdepth(S/I) = depth(S/I) under specified conditions

hdepth(S/I) = depth(S/I) for squarefree ideals with linear quotients

sdepth(S/I) ≥ depth(S/I) for ideals satisfying technical conditions

Abstract

We prove that if $I$ is a monomial ideal with linear quotients in a ring of polynomials $S$ in $n$ indeterminates and $\operatorname{depth}(S/I)=n-2$, then $\operatorname{sdepth}(S/I)=n-2$ and, if $I$ is squarefree, $\operatorname{hdepth}(S/I)=n-2$. Also, we prove that $\operatorname{sdepth}(S/I)\geq \operatorname{depth}(S/I)$ for a monomial ideal $I$ with linear quotients which satisfies certain technical conditions.