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

Abstract

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

11 pages; major revision - we corrected several proofs