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

S. A. Seyed Fakhari

• Let {\mathbb {K}} be a field and S={\mathbb {K}}[x_1,\ldots ,x_n] be the polynomial ring in n variables over {\mathbb {K}}. For any monomial ideal I, we denote its integral closure by {\overline{I}}. 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 Seyed Fakhari (J Algebra 489:463–474, 2017). 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\rightarrow \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 \begin{aligned} \mathrm{depth} (S/I^{sm}) \le \mathrm{depth} (S/{\overline{I}}), \end{aligned} 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).