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

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 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 \(\mathrm{Ass}(S/I)\subseteq \mathrm{Ass}(S/I^m)\).