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

Given an infinite field \(\mathbb{k}\) and a simplicial complex \(\Delta\), a common theme in studying the \(f\)- and \(h\)-vectors of \(\Delta\) has been the consideration of the Hilbert series of the Stanley--Reisner ring \(\mathbb{k}[\Delta]\) modulo a generic linear system of parameters \(\Theta\). 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 \(h_{d-1}^\mathfrak{a}(\Delta)\), the dimension over \(\mathbb{k}\) in degree \(d-1\) of \(\mathbb{k}[\Delta]/(\Theta)\), for any complex \(\Delta\) of dimension \(d-1\). In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series.