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Relative Poincaré duality in nonarchimedean geometry

Shizhang Li, Emanuel Reinecke, Bogdan Zavyalov

Abstract

We prove a conjecture of Bhatt-Hansen that derived pushforwards along proper morphisms of rigid-analytic spaces commute with Verdier duality on Zariski-constructible complexes. In particular, this yields duality statements for the intersection cohomology of proper rigid-analytic spaces. In our argument, we construct cycle classes in analytic geometry as well as trace maps for morphisms that are either smooth or proper or finite flat, with appropriate coefficients. As an application of our methods, we obtain new, significantly simplified proofs of $p$-adic Poincaré duality and the preservation of $\mathbf{F}_p$-local systems under smooth proper higher direct images.

Relative Poincaré duality in nonarchimedean geometry

Abstract

We prove a conjecture of Bhatt-Hansen that derived pushforwards along proper morphisms of rigid-analytic spaces commute with Verdier duality on Zariski-constructible complexes. In particular, this yields duality statements for the intersection cohomology of proper rigid-analytic spaces. In our argument, we construct cycle classes in analytic geometry as well as trace maps for morphisms that are either smooth or proper or finite flat, with appropriate coefficients. As an application of our methods, we obtain new, significantly simplified proofs of -adic Poincaré duality and the preservation of -local systems under smooth proper higher direct images.

Paper Structure

This paper contains 52 sections, 165 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Trace and duality for smooth maps
  4. Trace and duality for proper maps
  5. Preliminaries
  6. Valuation rings
  7. Curve-like affinoid fields
  8. Finite morphisms and residue fields
  9. Finite flat morphisms
  10. Trace for finite flat morphisms
  11. Cycle classes
  12. Cycle classes of divisors
  13. Construction of cycle classes
  14. First Chern classes
  15. Projective bundle and blow-up formulas
  16. ...and 37 more sections

Key Result

Theorem 1.1.1

Let $f\colon X \to Y$ be a proper morphism of rigid-analytic spaces over $K$, and let $\omega_X$ and $\omega_Y$ be the dualizing complexes on $X$ and $Y$ respectively. Then there is a canonical trace morphism $\mathop{\mathrm{Tr}}\nolimits_f \colon \mathrm{R} f_* \omega_X \to \omega_Y$ such that the is an equivalence for any $\mathcal{F} \in D_\mathrm{zc}(X_\mathrm{\acute{e}t}; \Lambda)$. In other

Theorems & Definitions (415)

  • Theorem 1.1.1: Bhatt--Hansen's conjecture, \ref{['general Poincare duality']}
  • Corollary 1.1.2: \ref{['cohomology duality']}
  • Corollary 1.1.3: Bhatt--Hansen's conjecture, \ref{['intersection cohomology duality']}
  • Theorem 1.2.1: \ref{['smooth-trace-constant']}, \ref{['Compatibility with algebraic geometry']}, \ref{['cor:compatibility-berkovich-trace-2']}, \ref{['lemma:comparison-LLZ']}
  • Remark 1.2.2
  • Remark 1.2.3
  • Theorem 1.2.4: \ref{['Poincare dualizability theorem']}, \ref{['general smooth Poincare duality']}
  • Corollary 1.2.5: \ref{['cor:preservation-lisse-sheaves']}
  • Theorem 1.3.1: Digest of \ref{['general proper trace subsection']}
  • Remark 1.3.2
  • ...and 405 more