Stanley depth of the integral closure of monomial ideals

Abstract

Let $$I$$ be a monomial ideal in the polynomial ring $$S=\mathbb K [x_1,\dots ,x_n]$$ . We study the Stanley depth of the integral closure $$\overline{I}$$ of $$I$$ . We prove that for every integer $$k\ge 1$$ , the inequalities $$\text{ sdepth} (S/\overline{I^k}) \le \text{ sdepth} (S/\overline{I})$$ and $$\text{ sdepth} (\overline{I^k}) \le \text{ sdepth} (\overline{I})$$ 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 $$\text{ sdepth} (S/I^{sk_1}) \le \text{ sdepth} (S/\overline{I})$$ and $$\text{ sdepth} (I^{sk_2}) \le \text{ sdepth} (\overline{I})$$ hold. In particular, $$\min _k \{\text{ sdepth} (S/I^k)\} \le \text{ sdepth} (S/\overline{I})$$ and $$\min _k \{\text{ sdepth} (I^k)\} \le \text{ sdepth} (\overline{I})$$ . We conjecture that for every integrally closed monomial ideal $$I$$ , the inequalities $$\text{ sdepth}(S/I)\ge n-\ell (I)$$ and $$\text{ sdepth} (I)\ge n-\ell (I)+1$$ 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.