Cohen-Macaulayness of large powers of Stanley-Reisner ideals

TL;DR

This paper characterizes when large symbolic and ordinary powers of Stanley-Reisner ideals are Cohen-Macaulay, linking these properties to matroids and complete intersections, and resolving open questions in the field.

Contribution

It provides a complete characterization of Cohen-Macaulayness of large powers of Stanley-Reisner ideals, connecting algebraic properties to combinatorial structures.

Findings

Symbolic powers are Cohen-Macaulay iff the complex is a matroid for m > 2.

Ordinary powers are Cohen-Macaulay for some m > 2 iff the complex is a complete intersection.

Results resolve several open questions on Cohen-Macaulayness of powers of Stanley-Reisner ideals.

Abstract

We prove that for m > 2, the m-th symbolic power of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex is a matroid. Similarly, the m-th ordinary power is Cohen-Macaulay for some m > 2 if and only if the complex is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.