Betti numbers of Stanley–Reisner rings with pure resolutions

Let ∆ be simplicial complex and let k[∆] denote the Stanley– Reisner ring corresponding to ∆. Suppose that k[∆] has a pure free resolution. Then we describe the Betti numbers and the Hilbert– Samuel multiplicity of k[∆] in terms of the h–vector of ∆. 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.