Betti numbers of Stanley--Reisner rings with pure resolutions

Details Betti numbers of Stanley--Reisner rings with pure resolutions Betti numbers of Stanley--Reisner rings with pure resolutions

2010-09-02

arxiv.org/abs/1009.0394v3

Mathematics

Combinatorics

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Scientific paper

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 and some inequalities for the components of the $h$--vector of the clique complex of an arbitrary chordal graph. As an other application, we derive a linear equation system and some inequalities for the components of the $h$--vector of Cohen--Macaulay simplicial complexes.

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