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

• S. A. Seyed Fakhari
Article

## Abstract

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

## Keywords

Depth Edge ideal Integral closure Stanley depth Stanley’s inequality

## Mathematics Subject Classification

13C15 05E40 13B22

## Notes

### Acknowledgements

The author thanks Irena Swanson for suggesting determinantal trick in the proof of Lemma 4.4. He also thanks the referee for careful reading of the paper and for valuable comments.

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