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On pristine morphisms

Javier Carvajal-Rojas, Axel Stäbler

Abstract

We investigate flat morphisms of schemes of positive characteristic whose relative Frobenius is an isomorphism, which we call pristine. We show that these give rise to a natural Grothendieck topology that is fine tuned for the localization of Cartier modules.

On pristine morphisms

Abstract

We investigate flat morphisms of schemes of positive characteristic whose relative Frobenius is an isomorphism, which we call pristine. We show that these give rise to a natural Grothendieck topology that is fine tuned for the localization of Cartier modules.

Paper Structure

This paper contains 8 sections, 23 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Generalities on Frobenius
  3. Regularity vs $F$-flatness
  4. Beyond the noetherian case
  5. On Pristinity
  6. Noetherian Pristinity
  7. Pristine Pullback of Cartier Modules
  8. Proof of \ref{['thm.TransRuleTestModulesPristinePullback']}

Key Result

Proposition 1

With notation as above, $F^e_{f}$ is an integral universal homeomorphism that induces purely inseparable residue field extensions.

Theorems & Definitions (77)

  • Proposition 1: stacks-project
  • Remark 1: Stability under base change and naturality
  • Definition 1: $F$-finiteness cf. HashimotoFfinitenessAndItsdescent
  • Remark 2
  • Remark 3: Kählerianity vs $F$-finiteness
  • Proposition 2
  • proof
  • Remark 4
  • Proposition 3
  • proof
  • ...and 67 more