Depth and Stanley depth of powers of the path ideal of a cycle graph

Abstract

Let $J_{n,m}:=(x_1{x}_2\cdots x_m,x_2{x}_3\cdots x_{m+1},\dots ,x_{n-m+1}\cdots x_n,x_{n-m+2}\cdots x_n{x}_1,\dots ,x_n{x}_1\cdots x_{m-1})$ be the $m$-path ideal of the cycle graph of length $n$, in the ring $S=K[x_1,\dots ,x_n]$. Let $d=\gcd (n,m)$. We prove that $\mathrm{depth}(S/J_{n,m}^t)\le d-1$ for all $t\ge n-1$. We show that $\mathrm{sdepth}(S/J_{n,n-1}^t)=\mathrm{depth}(S/J_{n,n-1}^t)=\max \{n-t-1,0\}$ for all $t\ge 1$. Also, we give some bounds for $\mathrm{depth}(S/J_{n,m}^t)$ and $\mathrm{sdepth}(S/J_{n,m}^t)$, where $t\ge 1$.

arXiv comment15 pages; correction of the main theorem