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On the Infinitely Generated Locus of Frobenius Algebras of Rings of Prime Characteristic
Alberto F. Boix
,
Danny A. J. Gómez–Ramírez
,
Santiago Zarzuela
Abstract
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.