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

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 $\mathrm{sdepth}(I(G{)}^k)\ge 2$, for any positive integer $k\le \mathrm{girth}(G)/2+1$. This result partially answers a question asked in [20]. For any proper monomial ideal $I$ of $S$, it is shown that the sequence $\{\mathrm{depth}(\overline{I^k}/\overline{I^{k+1}}){\}}_{k=0}^{\infty }$ is convergent and ${\lim }_{k\to \infty }\mathrm{depth}(\overline{I^k}/\overline{I^{k+1}})=n-\ell (I)$, 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 $$\mathrm{depth}(S/I^{sm})\le \mathrm{depth}(S/\overline{I}),$$for every integer $m\ge 1$. We also determine a value $s$ for which the above inequality holds. If $I$ is an integrally closed ideal, we show that $\mathrm{depth}(S/I^m)\le \mathrm{depth}(S/I)$, for every integer $m\ge 1$. As a consequence, we obtain that for any integrally closed monomial ideal $I$ and any integer $m\ge 1$, we have $\mathrm{Ass}(S/I)\subseteq \mathrm{Ass}(S/I^m)$. \end{abstract}

References in corpus (1)

• A non-partitionable Cohen-Macaulay simplicial complex