On the infinitely generated locus of Frobenius algebras of rings of prime characteristic (Q6563075)

Here is the main result of this paper:\N\NLet \(S = \mathbb{K}[x_{1}, \dots, x_{n}]/I \) be a Stanley-Reisner ring, where \(\mathbb{K}\) is a field of prime characteristic \(p\) and \(I\) is a square-free monomial ideal. Then the set\N\[\N\{\mathfrak{p} \in \mathrm{Spec}(S) : \mathcal{F}^{ E_{S_{p}}} \text{ is not finitely generated as a ring over its degree zero piece}\}\N\]\Nis closed. The authors determine the defining ideal of this closed set and provide an algorithmic method to compute it, both algebraic and combinatorial. Here, \( \mathcal{F}^{ E_{S_{p}}}\) is the Frobenius algebra of the injective hull of the residue field of the local ring \(S_{p}\). The Frobenius algebra is defined for a module \(M\) over a commutative Noetherian ring \(R\) of prime characteristic \(p\) as follows: \(\mathcal{F}^{ M}=\bigoplus_{e\ge0}\mathrm{End}_{p^{e}}(M)\), where \(\mathrm{End}_{p^{e}}(M)\) for \(e\ge0\) is the set of \(p^{e}\)-linear abelian group endomorphisms of \(M\), and multiplication is composition of maps.