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

TL;DR

This paper characterizes when symbolic and ordinary powers of Stanley-Reisner ideals are clean, linking this property to the combinatorial structure of the underlying simplicial complex, especially for matroids and dimension one cases.

Contribution

It establishes a precise equivalence between the cleanness of powers of Stanley-Reisner ideals and the matroid or Cohen-Macaulay properties of the simplicial complex.

Findings

$ riangle$ is a matroid iff $S/I_ riangle^{(m)}$ and $S/I_ riangle^m$ are clean for all $m$.

For $ ext{dim}( riangle)=1$, cleanness of $S/I_ riangle^{(2)}$ and $S/I_ riangle^2$ characterizes Cohen-Macaulayness.

The paper provides a combinatorial characterization of algebraic properties of ideal powers.

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.