Depth and Stanley depth of symbolic powers of cover ideals of graphs

Abstract

Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $J(G)$ is the cover ideal of $G$ and $J(G{)}^{(k)}$ is its $k$-th symbolic power. We prove that the sequences $\{\mathrm{sdepth}(S/J(G{)}^{(k)}){\}}_{k=1}^{\infty }$ and $\{\mathrm{sdepth}(J(G{)}^{(k)}){\}}_{k=1}^{\infty }$ are non-increasing and hence convergent. Suppose that ${\nu }_o(G)$ denotes the ordered matching number of $G$. We show that for every integer $k\ge 2{\nu }_o(G)-1$, the modules $J(G{)}^{(k)}$ and $S/J(G{)}^{(k)}$ satisfy the Stanley's inequality. We also provide an alternative proof for \cite[Theorem 3.4]{hktt} which states that $\mathrm{depth}(S/J(G{)}^{(k)})=n-{\nu }_o(G)-1$, for every integer $k\ge 2{\nu }_o(G)-1$.

arXiv commentarXiv admin note: text overlap with arXiv:1604.00656