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Mod $p$ Poincaré duality for $p$-adic period domains

Guillaume Pignon-Ywanne

TL;DR

The article develops a mod $p$ Poincaré duality theory for smooth partially proper rigid analytic varieties over a $p$-adic field by establishing a primitive comparison with compact support, and then applying Mann’s six functor formalism to relate compactly supported and usual étale cohomology. The authors prove that almost proper varieties and $p$-adic period domains are in the primitive comparison class, enabling PD statements with $\,\mathbb{F}_p$-coefficients and yielding explicit cohomology computations for Drinfeld spaces via generalized Steinberg representations. A central technical achievement is identifying Mann’s proper pushforward with Huber’s in the almost setting and proving PD on the essential image of the Riemann–Hilbert functor, which reduces PD for non-proper spaces to a finite hierarchy of localization and discretization steps. The results give a unified framework for PD across period domains and almost proper varieties, with extensions to arbitrary reductive groups and a detailed geometric decomposition à la Orlik that underpins the PD computations.

Abstract

In this article, we introduce a new class of smooth partially proper rigid analytic varieties over a $p$-adic field that satisfy Poincaré duality for étale cohomology with mod $p$-coefficients : the varieties satisfying "primitive comparison with compact support". We show that almost proper varieties, as well as p-adic (weakly admissible) period domains in the sense of Rappoport-Zink belong to this class. In particular, we recover Poincaré duality for almost proper varieties as first established by Li-Reinecke-Zavyalov, and we compute the étale cohomology with $\mathbb{F}_p$-coefficients of p-adic period domains, generalizing a computation of Colmez-Dospinescu-Niziol for Drinfeld's symmetric spaces. The arguments used in this paper rely crucially on Mann's six functors formalism for solid $\mathcal{O}^{+,a}/π$ coefficients.

Mod $p$ Poincaré duality for $p$-adic period domains

TL;DR

The article develops a mod Poincaré duality theory for smooth partially proper rigid analytic varieties over a -adic field by establishing a primitive comparison with compact support, and then applying Mann’s six functor formalism to relate compactly supported and usual étale cohomology. The authors prove that almost proper varieties and -adic period domains are in the primitive comparison class, enabling PD statements with -coefficients and yielding explicit cohomology computations for Drinfeld spaces via generalized Steinberg representations. A central technical achievement is identifying Mann’s proper pushforward with Huber’s in the almost setting and proving PD on the essential image of the Riemann–Hilbert functor, which reduces PD for non-proper spaces to a finite hierarchy of localization and discretization steps. The results give a unified framework for PD across period domains and almost proper varieties, with extensions to arbitrary reductive groups and a detailed geometric decomposition à la Orlik that underpins the PD computations.

Abstract

In this article, we introduce a new class of smooth partially proper rigid analytic varieties over a -adic field that satisfy Poincaré duality for étale cohomology with mod -coefficients : the varieties satisfying "primitive comparison with compact support". We show that almost proper varieties, as well as p-adic (weakly admissible) period domains in the sense of Rappoport-Zink belong to this class. In particular, we recover Poincaré duality for almost proper varieties as first established by Li-Reinecke-Zavyalov, and we compute the étale cohomology with -coefficients of p-adic period domains, generalizing a computation of Colmez-Dospinescu-Niziol for Drinfeld's symmetric spaces. The arguments used in this paper rely crucially on Mann's six functors formalism for solid coefficients.
Paper Structure (44 sections, 76 theorems, 135 equations)

This paper contains 44 sections, 76 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The proof of the main result
  4. Outline of the paper
  5. Conventions and notations
  6. Acknowledgements
  7. A Crash Course in Mann's six functor formalism
  8. The structure sheaf $\mathcal{O}^+/p$ for locally spatial diamonds
  9. Mann's functors $\mathcal{D}^a_{\square}(, \mathcal{O}^+/\pi)$ and $\mathcal{D}^a_{\square}(, \mathcal{O}^+/\pi)^\varphi$
  10. Classical $\mathbb{F}_p$-étale sheaves and the Riemann-Hilbert adjunction
  11. Poincaré Duality on the essential image of Riemann-Hilbert
  12. Standard almost $\mathcal{O}^+/\pi$-étale sheaves and discrete objects in $\mathcal{D}^a_{\square}(X, \mathcal{O}_X^+/\pi)$
  13. Standard almost $\mathcal{O}^+/\pi$-étale sheaves
  14. Identification of discrete objects
  15. About the topology of locally spatial diamonds
  16. ...and 29 more sections

Key Result

Theorem 1.1.1

(cf. PCCS_PD) Let $K$ be a complete extension of $\mathbb{Q}_p$, $\overline{K}$ be its algebraic closure, and $C$ be the completion of $\overline{K}$. Let $X$ be a partially proper proper rigid analytic variety over $K$ that is smooth of pure dimension $d$, and let $X_C$ be its base change to $C$. L is an almost isomorphism of $\mathcal{O}_C/p$-modules. When this is the case, we say that $X_C$ sat

Theorems & Definitions (185)

  • Theorem 1.1.1
  • Corollary 1.1.2
  • Proposition 1.1.3
  • Theorem 1.1.4
  • Corollary 1.1.5
  • Theorem 1.2.1
  • Theorem 1.2.2
  • Proposition 1.2.3
  • Lemma 2.1.1
  • proof
  • ...and 175 more