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On tame ramification and centers of $F$-purity

Javier Carvajal-Rojas, Anne Fayolle

Abstract

We introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves and sits nicely in between other notions of tame ramification in arithmetic geometry. However, when applied to the Frobenius map, it naturally yields the notion of center of $F$-purity (aka compatibly $F$-split subvariety). As an application, we describe the behavior of centers of $F$-purity under finite covers -- it all comes down to a transitivity property for tame ramification in towers.

On tame ramification and centers of $F$-purity

Abstract

We introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves and sits nicely in between other notions of tame ramification in arithmetic geometry. However, when applied to the Frobenius map, it naturally yields the notion of center of -purity (aka compatibly -split subvariety). As an application, we describe the behavior of centers of -purity under finite covers -- it all comes down to a transitivity property for tame ramification in towers.
Paper Structure (15 sections, 60 theorems, 46 equations)

This paper contains 15 sections, 60 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. A bit of history and background
  3. Overview of the paper
  4. Acknowledgements
  5. Criteria for Separability
  6. Transpositions of Ideals and Tame Ramification
  7. A workout example: diagonalizable quotients
  8. Cyclic case
  9. Tame ramification in separable normalizations
  10. Centers of $F$-Purity and the Cartier Core Map
  11. Fibered Homomorphisms
  12. Categorical formulation
  13. Fibered Transpositions
  14. On the work of Speyer
  15. Diagonalizable quotients, revisited

Key Result

Lemma 1

Let $L/K$ and $K'/K$ be field extensions inside a common field $L'$ such that $L/K$ and $L'/K'$ are finite. Let $\xi \colon L\otimes_K K' \xrightarrow{\ \ }L'$ be the canonical homomorphism. Then, $\xi$ is injective if and only if there is a $0\neq \lambda \in \mathop{\mathrm{Hom}}\nolimits_K(L,K)$

Theorems & Definitions (170)

  • Lemma 1
  • proof
  • Corollary 1: First criterion, cf. SchwedeTuckerTestIdealFiniteMaps
  • Corollary 2: Second criterion
  • proof
  • Definition 1: Transposition of ideals and degeneracy
  • Remark 1: On non-degeneracy
  • Proposition 1: Basic properties
  • proof
  • Proposition 2: Transitivity
  • ...and 160 more