Algebraic $h$-vectors of simplicial complexes through local cohomology, part 1

Abstract

Given an infinite field and a simplicial complex ∆, a common theme in studying the f -and hvectors of ∆ has been the consideration of the Hilbert series of the Stanley-Reisner ring [∆] modulo a generic linear system of parameters Θ. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of h a d−1 (∆), the dimension over in degree d − 1 of [∆]/(Θ), for any complex ∆ of dimension d − 1. In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series.