Stanley depth and symbolic powers of monomial ideals

TL;DR

This paper investigates the behavior of Stanley depth in symbolic powers of squarefree monomial ideals, establishing inequalities and limit behaviors, and introduces a new method for comparing Stanley depths of monomial ideal factors.

Contribution

It provides new inequalities and limit results for Stanley depth of symbolic powers of squarefree monomial ideals, and introduces a novel comparison method.

Findings

Stanley depth decreases with symbolic power scaling.

Limit behavior of Stanley depth analyzed for symbolic powers.

New method for comparing Stanley depths of monomial ideal factors.

Abstract

The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\geq 1$, the inequalities ${\rm sdepth} (S/I^{(ks)}) \leq {\rm sdepth} (S/I^{(s)})$ and ${\rm sdepth} (I^{(ks)}) \leq {\rm sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we show that for every integer $k\geq1$, ${\rm sdepth}(I^{(k+d)})\leq {\rm sdepth}(I^{{(k)}})$ and ${\rm sdepth}(S/I^{(k+d)})\leq {\rm sdepth}(S/I^{{(k)}})$. Finally, we consider the limit behavior of the Stanley depth of symbolic powers of a squarefree monomial ideal. We also introduce a method for comparing the Stanley depth of factors of monomial ideals.