Uniform bounds in excellent F_p-algebras and applications to semi-continuity

2025/03/18 by Lyu, Shiji
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.13846

Abstract

We study two important numerical invariants, Hilbert--Kunz multiplicity and F-signature, on the spectrum of a Noetherian F_p-algebra R that is not necessarily F-finite. When R is excellent, we show that the limits defining the invariants are uniform. As a consequence, we show that the F-signature is lower semi-continuous, and the Hilbert--Kunz multiplicity is upper semi-continuous provided R is locally equidimensional. Uniform convergence is achieved via a uniform version of Cohen--Gabber theorem. We prove the results under weaker conditions than excellence.

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