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arXiv · 2503.13846

Uniform bounds in excellent $\mathbf{F}_p$-algebras and applications to semi-continuity

Shiji Lyu
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Abstract

We study two important numerical invariants, Hilbert--Kunz multiplicity and $F$-signature, on the spectrum of a Noetherian $\mathbf{F}_p$-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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Shiji Lyu. 2025-03-18. Uniform bounds in excellent $\mathbf{F}_p$-algebras and applications to semi-continuity. https://arxiv.org/abs/2503.13846

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