Simplicial complexes of whisker type

TL;DR

This paper proves that certain simplicial complexes derived from zero-dimensional monomial ideals are shellable and vertex decomposable, and studies the linear resolutions and depth of associated Stanley--Reisner ideals and their powers.

Contribution

It provides a new explicit shelling and vertex decomposition for these complexes, and analyzes the linear resolutions and depth behavior of related ideals and their powers.

Findings

$ riangle(I)$ is shellable and vertex decomposable.

All powers of $L(I)$ have linear resolutions.

Depth of $L(I)^k$ equals $n$ for all $k \\geq n$.

Abstract

Let $I\subset K[x_1,\ldots,x_n]$ be a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that $\Delta(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $\Delta(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.