F-signature exists

TL;DR

This paper proves the existence of the F-signature, a numerical invariant that measures the asymptotic splitting behavior of Frobenius endomorphisms in Noetherian local rings of prime characteristic, extending previous special cases.

Contribution

It establishes the existence of the F-signature in general, develops uniform Hilbert-Kunz estimates, and analyzes its behavior under finite ring extensions.

Findings

Existence of F-signature for a broad class of rings.

Explicit formulas for F-signatures of finite quotient singularities.

Development of uniform Hilbert-Kunz estimates.

Abstract

Suppose R is a Noetherian local ring with prime characteristic p>0. In this article, we show the existence of a local numerical invariant, called the F-signature, which roughly characterizes the asymptotic growth of the number of splittings of the iterates of the Frobenius endomorphism of R. This invariant was first formally defined by C. Huneke and G. Leuschke and has previously been shown to exist only in special cases. The proof of our main result is based on the development of certain uniform Hilbert-Kunz estimates of independent interest. Additionally, we analyze the behavior of the F-signature under finite ring extensions and recover explicit formulae for the F-signatures of finite quotient singularities.