F-rationality of the ring of modular invariants

TL;DR

This paper characterizes when the ring of invariants under a finite group action in characteristic p>0 is F-rational, using Frobenius limits and dual F-signature, and provides examples distinguishing F-rationality from F-regularity.

Contribution

It introduces a new criterion for F-rationality of invariant rings based on Frobenius limits and dual F-signature, and constructs examples differentiating F-rationality from F-regularity.

Findings

Characterization of F-rational invariant rings using Frobenius limits.

Identification of conditions for positive dual F-signature in invariant rings.

Example of an invariant ring that is F-rational but not F-regular.

Abstract

Using the description of the Frobenius limit of modules over the ring of invariants under an action of a finite group on a polynomial ring over a field of characteristic $p>0$ developed by Symonds and the author, we give a characterization of the ring of invariants with a positive dual $F$-signature. Combining this result and Kemper's result on depths of the ring of invariants under an action of a permutation group, we give an example of an $F$-rational, but non-$F$-regular ring of invariants under the action of a finite group.