Continuity of Hilbert-Kunz multiplicity and F-signature

TL;DR

This paper proves that Hilbert-Kunz multiplicity and F-signature vary continuously in Cohen-Macaulay local rings of prime characteristic when parameters are perturbed within the m-adic topology.

Contribution

It establishes the continuity of Hilbert-Kunz multiplicity and F-signature functions in Cohen-Macaulay local rings, a new result in the understanding of their stability.

Findings

Proved continuity of Hilbert-Kunz multiplicity

Proved continuity of F-signature

Applicable to Cohen-Macaulay local rings in prime characteristic

Abstract

We establish the continuity of Hilbert-Kunz multiplicity and F-signature as functions from a Cohen-Macaulay local ring $(R,\m,k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\m$-adic topology.