Stability of depth and Stanley depth of symbolic powers of squarefree monomial ideals

Abstract

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 $I\subset S$ is a squarefree monomial ideal. For every integer $k\ge 1$, we denote the $k$-th symbolic power of $I$ by $I^{(k)}$. Recently, Montaño and Núñez-Betancourt \cite{mn} proved that for every pair of integers $m,k\ge 1$,$$\mathrm{depth}(S/I^{(m)})\le \mathrm{depth}(S/I^{(\lceil \frac{m}{k}\rceil )}).$$We provide an alternative proof for this inequality. Moreover, we reprove the known results that the sequence $\{\mathrm{depth}(S/I^{(k)}){\}}_{k=1}^{\infty }$ is convergent and$$\underset{k}{\min }\mathrm{depth}(S/I^{(k)})=\underset{k\to \infty }{\lim }\mathrm{depth}(S/I^{(k)})=n-{\ell }_s(I),$$where ${\ell }_s(I)$ denotes the symbolic analytic spread of $I$. We also determine an upper bound for the index of depth stability of symbolic powers of $I$. Next, we consider the Stanley depth of symbolic powers and prove that the sequences $\{\mathrm{sdepth}(S/I^{(k)}){\}}_{k=1}^{\infty }$ and $\{\mathrm{sdepth}(I^{(k)}){\}}_{k=1}^{\infty }$ are convergent and the limit of each sequence is equal to its minimum. Furthermore, we determine an upper bound for the indices of sdepth stability of symbolic powers.