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

Abstract:

Let $\mathbb {K}$ be a field and let $S=\mathbb {K}[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over $\mathbb {K}$. Assume that $I\subset S$ is a squarefree monomial ideal. For every integer $k\geq 1$, we denote the $k$-th symbolic power of $I$ by $I^{(k)}$. Recently, Montaño and Núñez-Betancourt (2018), and independently Nguyen and Trung (to appear), proved that for every pair of integers $k, i\geq 1$, \begin{equation*} \mathrm {depth}(S/I^{(k)})\leq \mathrm {depth}(S/I^{(\lceil \frac {k}{i}\rceil )}). \end{equation*} We provide an alternative proof for this inequality. Moreover, we re-prove the known results that the sequence $\{\mathrm {depth}(S/I^{(k)})\}_{k=1}^{\infty }$ is convergent and \begin{equation*} \min _k\mathrm {depth}(S/I^{(k)})=\lim _{k\rightarrow \infty }\mathrm {depth}(S/I^{(k)})=n-\ell _s(I), \end{equation*} 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.