arXiv2018
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}