On the behavior of $F$-signatures, splitting primes, and test modules under finite covers

TL;DR

This paper studies how $F$-signatures, splitting primes, and test modules change under finite covers, introducing the concept of transposability along sections of the relative canonical module to deepen understanding.

Contribution

It extends the notion of transposability along sections of the relative canonical module, providing a comprehensive framework for their behavior under finite covers.

Findings

Characterizes the behavior of $F$-signatures under finite covers

Introduces the concept of transposability along sections of the relative canonical module

Provides new insights into splitting primes and test modules in this context

Abstract

We give a comprehensive treatment on how $F$-signatures, splitting primes, splitting ratios, and test modules behave under finite covers. To this end, we expand on the notion of transposability along a section section of the relative canonical module as first introduced by K.~Schwede and K.~Tucker.