On the infinitely generated locus of Frobenius algebras of rings of prime characteristic

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. The main goal of this paper is to study in some detail when $$\overline{W^R}:=\{\mathfrak{p}\in \mathrm{Spec}(R):{\mathcal{F}}^{E_{\mathfrak{p}}}\text{ is finitely generated as a ring over its degree zero piece}\}$$ is an open set in the Zariski topology, where ${\mathcal{F}}^{E_{\mathfrak{p}}}$ denotes the Frobenius algebra attached to the injective hull of the residue field of $R_{\mathfrak{p}}\mathrm{.}$ We show that this is true when $R$ is a Stanley–Reisner ring; moreover, in this case, we explicitly compute its closed complement, providing an algorithmic method for doing so.