Stanley depth of the integral closure of monomial ideals

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 $k\ge 1$, the inequalities $\mathrm{sdepth}(S/\overline{I^k})\le \mathrm{sdepth}(S/\bar{I})$ and $\mathrm{sdepth}(\overline{I^k})\le \mathrm{sdepth}(\bar{I})$ hold. We also prove that for every monomial ideal $I\subset S$ there exist integers $k_1,k_2\ge 1$, such that for every $s\ge 1$, the inequalities $\mathrm{sdepth}(S/I^{s{k}_1})\le \mathrm{sdepth}(S/\bar{I})$ and $\mathrm{sdepth}(I^{s{k}_2})\le \mathrm{sdepth}(\bar{I})$ hold. In particular, ${\min }_k\{\mathrm{sdepth}(S/I^k)\}\le \mathrm{sdepth}(S/\bar{I})$ and ${\min }_k\{\mathrm{sdepth}(I^k)\}\le \mathrm{sdepth}(\bar{I})$. We conjecture that for every integrally closed monomial ideal $I$, the inequalities $\mathrm{sdepth}(S/I)\ge n-\ell (I)$ and $\mathrm{sdepth}(I)\ge n-\ell (I)+1$ 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.

References in corpus (1)

• Integral closure of ideals