The cleanness of (symbolic) powers of Stanley-Reisner ideals

Abstract

Let $\Delta$ be a pure simplicial complex and $I_{\Delta }$ its Stanley-Reisner ideal in a polynomial ring $S$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_{\Delta }^{(m)}$ ($S/I_{\Delta }^m$) is clean for all $m\in \mathbb{N}$. If $\dim (\Delta )=1$, we also prove that $S/I_{\Delta }^{(2)}$ ($S/I_{\Delta }^2$) is clean if and only if $S/I_{\Delta }^{(2)}$ ($S/I_{\Delta }^2$) is Cohen-Macaulay.