Rings of Frobenius operators

TL;DR

This paper investigates the structure of Frobenius operator rings over local rings of prime characteristic, revealing conditions for finite generation and applications to F-jumping numbers in determinantal rings.

Contribution

It provides new insights into the finite generation of Frobenius operator rings and concrete descriptions in determinantal rings, advancing understanding of their algebraic properties.

Findings

F(E) may not be finitely generated over its degree zero part in determinantal rings.

Concrete descriptions of F(E) are obtained in general settings.

F-jumping numbers are shown to be discrete in determinantal rings.

Abstract

Let R be a local ring of prime characteristic. We study the ring of Frobenius operators F(E), where E is the injective hull of the residue field of R. In particular, we examine the finite generation of F(E) over its degree zero component, and show that F(E) need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of F(E) in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in determinantal rings.