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

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