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

Abstract

Given an infinite field k and a simplicial complex Δ, a common theme in studying the f- and h-vectors of Δ has been the consideration of the Hilbert series of the Stanley–Reisner ring k[Δ] 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 hd−1a(Δ), the dimension over k in degree d−1 of k[Δ]/(Θ), for any complex Δ of dimension d−1. In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series.