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

$$ \{\mathfrak {p}\in {\text {Spec}} (R)\,:\, \mathcal {F}^{E_{\mathfrak {p}}}\text { is finitely generated as a ring over its degree zero piece}\} $$

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 \(R_{\mathfrak {p}}.\) 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.