Glicci simplicial complexes

TL;DR

This paper proves that Stanley-Reisner ideals of weakly vertex-decomposable simplicial complexes are glicci, advancing liaison theory, and shows that glicci property can depend on the field characteristic, with implications for Stanley's conjectures.

Contribution

It establishes that certain classes of simplicial complexes have glicci Stanley-Reisner ideals and demonstrates the characteristic dependence of the glicci property.

Findings

Weakly vertex-decomposable complexes are glicci.

Glicci property depends on the characteristic of the base field.

Provides evidence for Stanley's conjectures on partitionable complexes.

Abstract

One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this question for Stanley-Reisner ideals defined by simplicial complexes that are weakly vertex-decomposable. This class of complexes includes matroid, shifted and Gorenstein complexes respectively. Moreover, we construct a simplicial complex which shows that the property of being glicci depends on the characteristic of the base field. As an application of our methods we establish new evidence for two conjectures of Stanley on partitionable complexes and on Stanley decompositions.