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Mod-p Poincaré Duality in p-adic Analytic Geometry

Bogdan Zavyalov

Abstract

We show Poincaré Duality for $\mathbf{F}_p$-étale cohomology of a smooth proper rigid-analytic space over a non-archimedean field $K$ of mixed characteristic $(0, p)$. It positively answers the question raised by P. Scholze in [Sch13a]. We prove duality via constructing Faltings' trace map relating Poincaré Duality on the generic fiber to (almost) Grothendieck Duality on the mod-$p$ fiber of a formal model. We also formally deduce Poincaré Duality for $\mathbf{Z}/p^n\mathbf{Z}$, $\mathbf{Z}_p$, and $\mathbf{Q}_p$-coefficients.

Mod-p Poincaré Duality in p-adic Analytic Geometry

Abstract

We show Poincaré Duality for -étale cohomology of a smooth proper rigid-analytic space over a non-archimedean field of mixed characteristic . It positively answers the question raised by P. Scholze in [Sch13a]. We prove duality via constructing Faltings' trace map relating Poincaré Duality on the generic fiber to (almost) Grothendieck Duality on the mod- fiber of a formal model. We also formally deduce Poincaré Duality for , , and -coefficients.

Paper Structure

This paper contains 47 sections, 135 theorems, 423 equations.

Table of Contents

  1. Introduction
  2. Historical overview
  3. Main steps of our proof
  4. Acknowledgements
  5. Conventions
  6. Poor man's version of duality on formal schemes
  7. Finitely presented compactifications
  8. Grothendieck duality for universally coherent schemes
  9. Reflexivity of the dualizing sheaf
  10. Dualizing modules on separated admissible formal schemes
  11. Extension properties of dualizing sheaf
  12. Trace map for proper morphisms of separated admissible formal schemes
  13. Dualizing complexes on smooth, separated rigid-analytic spaces
  14. Faltings' Trace Map
  15. Idea of the construction
  16. ...and 32 more sections

Key Result

Theorem 1.1.1

(Classical Poincaré Duality) Let $X$ be a compact complex manifold $X$ of pure dimension $d$. Then the pairing is perfect for any $i\geq 0$.

Theorems & Definitions (362)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Theorem 1.1.3
  • Theorem 1.1.4
  • Remark 1.1.5
  • Definition 2.1.1
  • Remark 2.1.2
  • Remark 2.1.3
  • Lemma 2.1.4
  • proof
  • ...and 352 more