On the depth and Stanley depth of integral closure of powers of monomial ideals

TL;DR

This paper investigates the depth and Stanley depth of powers and integral closures of monomial ideals, especially edge ideals of graphs, establishing inequalities, convergence of depth sequences, and properties of associated primes.

Contribution

It proves Stanley's inequality for integral closures of powers of edge ideals and explores depth behavior of monomial ideals and their integral closures, providing new bounds and convergence results.

Findings

Stanley's inequality holds for modules involving integral closures of edge ideals.

Depth sequences of integral closures converge to a specific limit related to analytic spread.

For integrally closed ideals, depth inequalities and associated prime inclusions are established.

Abstract

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $G$ is a graph with edge ideal $I(G)$. We prove that the modules $S/\overline{I(G)^k}$ and $\overline{I(G)^k}/\overline{I(G)^{k+1}}$ satisfy Stanley's inequality for every integer $k\gg 0$. If $G$ is a non-bipartite graph, we show that the ideals $\overline{I(G)^k}$ satisfy Stanley's inequality for all $k\gg 0$. For every connected bipartite graph $G$ (with at least one edge), we prove that ${\rm sdepth}(I(G)^k)\geq 2$, for any positive integer $k\leq {\rm girth}(G)/2+1$. This result partially answers a question asked in [20]. For any proper monomial ideal $I$ of $S$, it is shown that the sequence $\{{\rm depth}(\overline{I^k}/\overline{I^{k+1}})\}_{k=0}^{\infty}$ is convergent and $\lim_{k\rightarrow\infty}{\rm depth}(\overline{I^k}/\overline{I^{k+1}})=n-\ell(I)$, where $\ell(I)$ denotes the analytic spread of $I$. Furthermore, it is proved that for any monomial ideal $I$, there exists an integer $s$ such that $${\rm depth} (S/I^{sm}) \leq {\rm depth} (S/\overline{I}),$$for every integer $m\geq 1$. We also determine a value $s$ for which the above inequality holds. If $I$ is an integrally closed ideal, we show that ${\rm depth}(S/I^m)\leq {\rm depth}(S/I)$, for every integer $m\geq 1$. As a consequence, we obtain that for any integrally closed monomial ideal $I$ and any integer $m\geq 1$, we have ${\rm Ass}(S/I)\subseteq {\rm Ass}(S/I^m)$. \end{abstract}