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Cohomologically smooth morphisms for étale $\mathbb F_p$-sheaves in characteristic $p$

Felix Lotter

Abstract

We scrutinise the notions of cohomologically smooth morphisms and smooth objects for the six functor formalism of étale $\mathbb F_p$-sheaves on schemes in characteristic $p$. We show that only cohomologically étale morphisms are cohomologically smooth in this setting. This is complemented by a characterisation of cohomologically étale morphisms in arbitrary characteristic. In fact, we prove that such a morphism is already étale up to universal homeomorphism.

Cohomologically smooth morphisms for étale $\mathbb F_p$-sheaves in characteristic $p$

Abstract

We scrutinise the notions of cohomologically smooth morphisms and smooth objects for the six functor formalism of étale -sheaves on schemes in characteristic . We show that only cohomologically étale morphisms are cohomologically smooth in this setting. This is complemented by a characterisation of cohomologically étale morphisms in arbitrary characteristic. In fact, we prove that such a morphism is already étale up to universal homeomorphism.
Paper Structure (23 sections, 95 theorems, 49 equations)

This paper contains 23 sections, 95 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. The 6-functor formalism
  3. Cohomologically smooth morphisms
  4. Cohomologically étale morphisms
  5. Smooth objects
  6. Relation to the Riemann-Hilbert correspondence
  7. Overview
  8. Acknowledgements
  9. Abstract 6-functor formalisms
  10. The definition
  11. Construction of 6-functor formalisms
  12. The 6 functors for étale sheaves
  13. Cohomologically smooth morphisms
  14. Smooth and proper objects
  15. Properties of cohomologically smooth morphisms
  16. ...and 8 more sections

Key Result

Theorem 1

Let $k$ be a field of characteristic $p$ and let $\mathcal{D}_p$ be the 6-functor formalism of étale $\mathbb F_p$-sheavesSee thm:et 6 functors for the precise definition. on the category $\mathcal{C}$ of finite type, separated schemes over $k$. Let $f:Y \to X$ be a morphism in $\mathcal{C}$. Then t where $f^\mathrm{perf}$ denotes the perfection of $f$ (see prop:perf and awn).

Theorems & Definitions (228)

  • Theorem : \ref{['thm:char']}
  • Theorem : \ref{['cor:coh sm quasifinite coh etale']}
  • Theorem : \ref{['cor:supp of sm coh et']}
  • Definition 2.1: Mann
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5: Mann
  • Proposition 2.7
  • proof
  • ...and 218 more