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.