Betti numbers of Stanley--Reisner rings with pure resolutions

Abstract

Let $\Delta$ be simplicial complex and let $k[\Delta ]$ denote the Stanley--Reisner ring corresponding to $\Delta$. Suppose that $k[\Delta ]$ has a pure free resolution. Then we describe the Betti numbers and the Hilbert--Samuel multiplicity of $k[\Delta ]$ in terms of the $h$--vector of $\Delta$. As an application, we derive a linear equation system for the components of the $h$--vector of the clique complex of an arbitrary chordal graph.

10 pages, new formulas for the components of the clique complex of chordal graphs