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Poincaré duality for rigid analytic Hyodo--Kato theory

Veronika Ertl, Kazuki Yamada

Abstract

The purpose of this paper is to establish Hyodo--Kato theory with compact support for semistable schemes through rigid analytic methods. To this end we introduce several types of log rigid cohomology with compact support. moreover we show that additional structures on the (rigid) Hyodo--Kato cohomology and the Hyodo--Kato map introduced in our previous paper are compatible with Poincaré duality. Compared to the crystalline approach, the constructions are explicit yet versatile, and hence suitable for computations.

Poincaré duality for rigid analytic Hyodo--Kato theory

Abstract

The purpose of this paper is to establish Hyodo--Kato theory with compact support for semistable schemes through rigid analytic methods. To this end we introduce several types of log rigid cohomology with compact support. moreover we show that additional structures on the (rigid) Hyodo--Kato cohomology and the Hyodo--Kato map introduced in our previous paper are compatible with Poincaré duality. Compared to the crystalline approach, the constructions are explicit yet versatile, and hence suitable for computations.

Paper Structure

This paper contains 8 sections, 44 theorems, 201 equations.

Table of Contents

  1. Log overconvergent isocrystals
  2. Log overconvergent isocrystals associated to log substructures
  3. Log rigid cohomology with compact support
  4. Crystalline cohomology with compact support revisited
  5. Comparison of compactly supported log crystalline and log rigid cohomology
  6. Poincaré duality
  7. Extension groups of $(\varphi,N)$-modules
  8. The cup product on Hyodo--Kato cohomology

Key Result

Theorem \oldthetheorem

Let $Y$ be a proper strictly semistable log scheme over $k^0$ and $\mathscr{E}$ a unipotent log overconvergent isocrystal on $Y$ over $W^\varnothing$. There exists a diagram of natural morphisms \xymatrix{ \ar@{}[rd]|>>>>\circlearrowright& R\Gamma_{{\mathrm{rig}},c}(Y/W^\varnothing,\mathscr{E})\ar[ Here $F\{s\}$ denotes the global section of the generic fiber of $\mathcal{S}$, that is the ring of

Theorems & Definitions (132)

  • Theorem \oldthetheorem: Corollary \ref{['cor: compare rigid']}
  • Theorem \oldthetheorem: Theorem \ref{['thm: dual']}
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 122 more