Betti numbers of Stanley--Reisner rings with pure resolutions

TL;DR

This paper characterizes Betti numbers and multiplicity of Stanley--Reisner rings with pure resolutions using the h-vector of the simplicial complex, providing new algebraic-combinatorial relations and applications to chordal graphs.

Contribution

It offers explicit formulas for Betti numbers and multiplicity of pure-resolution Stanley--Reisner rings based on the h-vector, and applies these results to chordal graph clique complexes.

Findings

Derived formulas linking Betti numbers and h-vector.

Established a linear system for h-vector components.

Applied results to chordal graph clique complexes.

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.