The face ideal of a simplicial complex

TL;DR

This paper introduces the face ideal of a simplicial complex, explores its algebraic properties, and applies the concept to poset ideals, showing that certain complexes are shellable.

Contribution

It defines the face ideal with linear quotients, relates it to whisker complexes via Alexander duality, and introduces higher dimensional whisker complexes with shellable independence complexes.

Findings

Face ideal has linear quotients.

Alexander dual of the face ideal is a whisker complex.

Higher dimensional whisker complexes are shellable.

Abstract

Given a simplicial complex we associate to it a squarefree monomial ideal which we call the face ideal of the simplicial complex, and show that it has linear quotients. It turns out that its Alexander dual is a whisker complex. We apply this construction in particular to chain and antichain ideals of a finite partially ordered set. We also introduce so-called higher dimensional whisker complexes and show that their independence complexes are shellable.