The asymptotic behavior of Frobenius direct images of rings of invariants

TL;DR

This paper introduces the Frobenius limit for modules over rings of prime characteristic, computes it for modules over invariant rings under finite group actions, and generalizes the generalized F-signature to the modular case.

Contribution

It defines the Frobenius limit in a new context and calculates it explicitly for modules over invariant rings, extending previous results to the modular setting.

Findings

Calculated Frobenius limits for modules over invariant rings under finite group actions.

Extended the generalized F-signature description to the modular case.

Provided explicit formulas for Frobenius limits in the context of invariant rings.

Abstract

We define the Frobenius limit of a module over a ring of prime characteristic to be the limit of the normalized Frobenius direct images in a certain Grothendieck group. When a finite group acts on a polynomial ring, we calculate this limit for all the modules over the twisted group algebra that are free over the polynomial ring; we also calculate the Frobenius limit for the restriction of these to the ring of invariants. As an application, we generalize the description of the generalized $F$-signature of a ring of invariants by the second author and Nakajima to the modular case.