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

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.