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

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 h d − 1 a (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. [ABSTRACT FROM AUTHOR]