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

TL;DR

This paper investigates the conditions under which the locus of primes with finitely generated Frobenius algebras is open in the spectrum of a prime characteristic ring, with explicit results for Stanley--Reisner rings.

Contribution

It characterizes the openness of the Frobenius algebra locus in prime characteristic rings and provides an explicit computation method for Stanley--Reisner rings.

Findings

The Frobenius algebra locus is open for Stanley--Reisner rings.

An explicit description of the closed complement of this locus is given.

An algorithmic approach to compute the closed complement is developed.

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 \[ \overline{W^R}:=\{\mathfrak{p}\in\operatorname{Spec} (R):\ \mathcal{F}^{E_{\mathfrak{p}}}\text{ is finitely generated as a ring over its degree zero piece}\} \] 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.