Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Shiji Lyu

TL;DR

The paper establishes uniform convergence for the defining limits of the Hilbert--Kunz multiplicity $e_{\mathrm{HK}}$ and the $F$-signature $s$ over spectra of Noetherian $\mathbf{F}_p$-algebras not necessarily $F$-finite, under (quasi-)excellence assumptions and weaker refinements. It develops a uniform Cohen--Gabber framework to bound discriminants and ramification, combining taming techniques with a localization-completion approach to obtain global uniform bounds from local data. Consequently, it proves that $\mathfrak{p}\mapsto s(R_{\mathfrak{p}})$ is lower semi-continuous and, when $R$ is locally equidimensional, $\mathfrak{p}\mapsto e_{\mathrm{HK}}(R_{\mathfrak{p}})$ is upper semi-continuous, with both invariants expressible as uniform limits of constructible, semi-continuous families. The work also provides a non-complete Cohen--Gabber variant and discusses extensions to $F$-rational and paired settings, as well as potential connections to rigid analytic geometry and broader questions about fixed finite Frobenius-like maps. Overall, the results advance the understanding of $F$-singularities and their numerical invariants in more general algebraic settings and offer tools for studying their behavior in families.

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.

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

TL;DR

The paper establishes uniform convergence for the defining limits of the Hilbert--Kunz multiplicity and the -signature over spectra of Noetherian -algebras not necessarily -finite, under (quasi-)excellence assumptions and weaker refinements. It develops a uniform Cohen--Gabber framework to bound discriminants and ramification, combining taming techniques with a localization-completion approach to obtain global uniform bounds from local data. Consequently, it proves that is lower semi-continuous and, when is locally equidimensional, is upper semi-continuous, with both invariants expressible as uniform limits of constructible, semi-continuous families. The work also provides a non-complete Cohen--Gabber variant and discusses extensions to -rational and paired settings, as well as potential connections to rigid analytic geometry and broader questions about fixed finite Frobenius-like maps. Overall, the results advance the understanding of -singularities and their numerical invariants in more general algebraic settings and offer tools for studying their behavior in families.

Abstract

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

Paper Structure

This paper contains 32 sections, 53 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. The subject
  3. Summary of results
  4. The strategy
  5. Further directions
  6. Acknowledgment
  7. Preliminaries
  8. Excellence conditions
  9. Notions of $F$-singularities
  10. Local Bertini
  11. Discriminant
  12. A non-complete version of Cohen--Gabber
  13. Tame ramification
  14. A condition of one-dimensional local rings
  15. Tame curves
  16. ...and 17 more sections

Key Result

Theorem 1.2.1

Assume that $R$ is $(R_0)$. Then there exist constants $\delta,\mu,\Delta\in\mathbf{Z}_{\geq 0}$ depending only on $R$, and a quasi-finite, syntomic ring map $R\to S$, such that for all $\mathfrak{p}\in\operatorname{Spec}(R)$, there exist a $\mathfrak{q}\in\operatorname{Spec}(S)$ above $\mathfrak{p}

Theorems & Definitions (108)

  • Theorem 1.2.1: a uniform version of Cohen--Gabber theorem; see Theorem \ref{['thm:UnifCohenGabber']}
  • Theorem 1.2.2: see Theorem \ref{['thm:unifbound']}
  • Theorem 1.2.3: see Theorem \ref{['thm:eHKSemicont']}
  • Theorem 1.2.4: see Theorem \ref{['thm:sSemicont']} and Corollary \ref{['cor:sFRlocusopen']}
  • Theorem 1.2.5: see Lemma \ref{['lem:individualHKconstr']} and Corollary \ref{['cor:individualHKsemicont']}
  • Theorem 1.2.6: see Lemma \ref{['lem:SeSemicont']}
  • Theorem 2.3.1: Flenner-local-Bertini
  • Lemma 2.3.2
  • proof
  • Definition 2.4.1
  • ...and 98 more