Equality of ordinary and symbolic powers of Stanley-Reisner ideals

TL;DR

This paper characterizes simplicial complexes where the m-th symbolic and ordinary powers of Stanley-Reisner ideals coincide, providing finite classifications and implications for Cohen-Macaulay properties in two-dimensional cases.

Contribution

It offers the first combinatorial characterizations of such complexes and fully describes all complexes with this property in two dimensions.

Findings

Finite classification of complexes with equal symbolic and ordinary powers

Complete description of these complexes in two dimensions

Identification of complexes with Cohen-Macaulay powers for given m

Abstract

This paper studies properties of simplicial complexes for which the m-th symbolic power of the Stanley-Reisner ideal equals to the m-th ordinary power for a given m > 1. The main results are combinatorial characterizations of such complexes in the two-dimensional case. It turns out that there exist only a finite number of complexes with this property and that these complexes can be described completely. As a consequence we are able to determine all complexes for which the m-th ordinary power of the Stanley-Reisner ideal is Cohen-Macaulay for a given m > 1.