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Etale cohomology of diamonds

Peter Scholze

Abstract

Motivated by problems on the étale cohomology of Rapoport--Zink spaces and their generalizations, as well as Fargues's geometrization conjecture for the local Langlands correspondence, we develop a six functor formalism for the étale cohomology of diamonds, and more generally small v-stacks on the category of perfectoid spaces of characteristic $p$. Using a natural functor from analytic adic spaces over $\mathbb Z_p$ to diamonds which identifies étale sites, this induces a similar formalism in that setting, which in the noetherian setting recovers the formalism from Huber's book.

Etale cohomology of diamonds

Abstract

Motivated by problems on the étale cohomology of Rapoport--Zink spaces and their generalizations, as well as Fargues's geometrization conjecture for the local Langlands correspondence, we develop a six functor formalism for the étale cohomology of diamonds, and more generally small v-stacks on the category of perfectoid spaces of characteristic . Using a natural functor from analytic adic spaces over to diamonds which identifies étale sites, this induces a similar formalism in that setting, which in the noetherian setting recovers the formalism from Huber's book.

Paper Structure

This paper contains 27 sections, 243 theorems, 453 equations.

Table of Contents

  1. Introduction
  2. Spectral Spaces
  3. Perfectoid Spaces
  4. Set-theoretic bounds
  5. Morphisms of perfectoid spaces
  6. Étale morphisms
  7. Totally disconnected spaces
  8. The pro-étale and v-topology
  9. Descent
  10. Morphisms of v-stacks
  11. Diamonds
  12. Small v-stacks
  13. Spatial morphisms
  14. Comparison of étale, pro-étale and v-cohomology
  15. Analytic adic spaces as diamonds
  16. ...and 12 more sections

Key Result

Theorem 1.2

The v-topology on $\mathop{\mathrm{Perf}}\nolimits$ is subcanonical, and for any affinoid perfectoid space $X=\mathop{\mathrm{Spa}}\nolimits(R,R^+)$, $H^0_v(X,\mathcal{O}_X) = R$, $H^0_v(X,\mathcal{O}_X^+)=R^+$ and for $i>0$, $H^i_v(X,\mathcal{O}_X)=0$ and $H^i_v(X,\mathcal{O}_X^+)$ is almost zero.

Theorems & Definitions (343)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 333 more