F-signature and Hilbert-Kunz Multipicity: a combined approach and comparison

TL;DR

This paper introduces a unified framework for studying Hilbert-Kunz multiplicity and F-signature in positive characteristic algebra, providing simplified proofs and new insights into their properties and relationships.

Contribution

It presents a combined approach that simplifies proofs and offers new perspectives on the connection between F-signature and Hilbert-Kunz multiplicity.

Findings

Simplified proofs of existence, semicontinuity, and positivity.

F-signature characterized as infimum of Hilbert-Kunz differences.

Affirmative answer to Watanabe and Yoshida's question.

Abstract

We present a unified approach to the study of Hilbert-Kunz multiplicity, F-signature, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question of Watanabe and Yoshida allowing the F-signature to be viewed as the infimum of relative differences in the Hilbert-Kunz multiplicites of the cofinite ideals in a local ring.