Given an infinite field $\mathbb{k}$ 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 $\mathbb{k}[Δ]$ 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 $h_{d-1}^\mathfrak{a}(Δ)$, the dimension over $\mathbb{k}$ in degree $d-1$ of $\mathbb{k}[Δ]/(Θ)$, 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.