Uniform Bounds in F-Finite Rings and Lower Semi-Continuity of the F-Signature

TL;DR

This paper proves uniform bounds in F-finite rings, demonstrating the lower semi-continuity of the F-signature function and providing a new proof for the upper semi-continuity of Hilbert-Kunz multiplicity, advancing understanding of these invariants.

Contribution

It introduces uniform bounds in F-finite rings and proves semi-continuity properties of F-signature and Hilbert-Kunz multiplicity, with a new proof for the latter's upper semi-continuity.

Findings

F-signature function is lower semi-continuous.

Hilbert-Kunz multiplicity is upper semi-continuous.

Uniform convergence of length functions to their limits.

Abstract

This paper establishes uniform bounds in characteristic $p$ rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the Spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the F-signature function. From this we establish that the F-signature function is lower semi-continuous. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, which was originally proven by Ilya Smirnov.