On the k-Buchsbaum property of powers of Stanley–Reisner ideals

Abstract

Let $S=K[x_1,x_2,\dots ,x_n]$ be a polynomial ring over a field $K$. Let $\Delta$ be a simplicial complex whose vertex set is contained in $\{1,2,\dots ,n\}$. For an integer $k\ge 0$, we investigate the $k$-Buchsbaum property of residue class rings $S/I^{\left(t\right)}$ and $S/I^t$ for the Stanley–Reisner ideal $I=I_{\Delta }$. We characterize the $k$-Buchsbaumness of such rings in terms of the simplicial complex $\Delta$ and the power $t$. We also give a characterization in the case where $I$ is the edge ideal of a simple graph.