Simplicial complexes of Whisker type

Mina Bigdeli ; Jürgen Herzog ; Takayuki Hibi ; Antonio Macchia

The electronic journal of combinatorics, Tome 22 (2015) no. 1

Résumé

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 $k\geq n$.

Détail

DOI : 10.37236/4894

Classification : 13C15, 05E40, 05E45, 13D02
Mots-clés : depth function, linear quotients, vertex decomposable, Whisker complexes, zero-dimensional ideals