Stanley depth of the integral closure of monomial ideals

TL;DR

This paper investigates the Stanley depth of the integral closure of monomial ideals, establishing inequalities and conjecturing bounds related to the analytic spread, with implications for Stanley's conjecture in large powers.

Contribution

It proves inequalities for Stanley depth of integral closures and their powers, and proposes a conjecture linking Stanley depth to the analytic spread for integrally closed monomial ideals.

Findings

Inequalities ${ m sdepth} (S/ar{I^k}) \,\leq\, {\rm sdepth} (S/\bar{I})$ and ${\rm sdepth} (\bar{I^k}) \,\leq\, {\rm sdepth} (\bar{I})$ hold.

Existence of integers $k_1,k_2$ such that ${\rm sdepth} (S/I^{sk_1}) \,\leq\, {\rm sdepth} (S/\bar{I})$ and ${\rm sdepth} (I^{sk_2}) \,\leq\, {\rm sdepth} (\bar{I})$.

Conjecture that Stanley depth bounds relate to the analytic spread, implying Stanley's conjecture for large powers if the ideal is normal.

Abstract

Let $I$ be a monomial ideal in the polynomial ring $S=\mathbb{K}[x_1,...,x_n]$. We study the Stanley depth of the integral closure $\bar{I}$ of $I$. We prove that for every integer $k\geq 1$, the inequalities ${\rm sdepth} (S/\bar{I^k}) \leq {\rm sdepth} (S/\bar{I})$ and ${\rm sdepth} (\bar{I^k}) \leq {\rm sdepth} (\bar{I})$ hold. We also prove that for every monomial ideal $I\subset S$ there exist integers $k_1,k_2\geq 1$, such that for every $s\geq 1$, the inequalities ${\rm sdepth} (S/I^{sk_1}) \leq {\rm sdepth} (S/\bar{I})$ and ${\rm sdepth} (I^{sk_2}) \leq {\rm sdepth} (\bar{I})$ hold. In particular, $\min_k \{{\rm sdepth} (S/I^k)\} \leq {\rm sdepth} (S/\bar{I})$ and $\min_k \{{\rm sdepth} (I^k)\} \leq {\rm sdepth} (\bar{I})$. We conjecture that for every integrally closed monomial ideal $I$, the inequalities ${\rm sdepth}(S/I)\geq n-\ell(I)$ and ${\rm sdepth} (I)\geq n-\ell(I)+1$ hold, where $\ell(I)$ is the analytic spread of $I$. Assuming the conjecture is true, it follows together with the Burch's inequality that Stanley's conjecture holds for $I^k$ and $S/I^k$ for $k\gg 0$, provided that $I$ is a normal ideal.