On the Topological Structure of the (Non-) Finitely-generated Locus of Frobenius Algebras emerging from Stanley-Reisner Rings

TL;DR

This paper investigates the topological properties of the locus where Frobenius algebras are finitely generated in Stanley-Reisner rings, providing partial answers to a conjecture about its openness and demonstrating non-trivial examples.

Contribution

It offers new insights into the topological structure of the finitely-generated locus of Frobenius algebras in Stanley-Reisner rings, including partial validation of a conjecture and explicit examples.

Findings

The finitely-generated locus has non-empty interior in certain Stanley-Reisner rings.

The locus contains open sets and its complement intersects open and closed sets.

Explicit examples show the locus is a non-trivial open set.

Abstract

In this paper we study initial topological properties of the (non-)finitely-generated locus of Frobenius Algebra coming from Stanley-Reisner rings defined through face ideals. More specifically, we will give a partial answer to a conjecture made by M. Katzman about the openness of the finitely generated locus of such Frobenius algebras. This conjecture can be formulated in a precise manner: Let us define \begin{equation*} U=\{P\in \text{Spec}(R)~:~\mathcal{F}(E_{R_{p}})~\text{is a finitely generated}~R_{p}\text{-algebra}\} \end{equation*} where $\mathcal{F}$ denotes the Frobenius functor and $E_{R_{p}}$ the injective hull of the residual field of the local ring $(R_{p},pR_{p})$. Is the locus $U$ an open set in the Zariski Topology? In the case where $R$ is a ring of the form $R=K[[x_{1},\dots ,x_{n}]]/I,$ where $I\subset R$ is a face ideal, i.e., square-free monomial ideal, we show that the corresponding $U$ has non-empty interior. Even more, we prove that in general $U$ contains two different types of opens sets and that in specific situation its complement contains intersections of opens and closes sets in the Zariski topology. Furthermore, we explicitly verify in some non-trivial examples that $U$ is an non-trivial open set.