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Poincaré duality in logarithmic motivic homotopy theory

Doosung Park

Abstract

By adapting arguments of Annala-Hoyois-Iwasa in the log setting, we prove Poincaré duality for smooth projective morphisms in logarithmic motivic homotopy theory. As an application, we show that the crystalline cohomology of a log compactification is independent of the choice.

Poincaré duality in logarithmic motivic homotopy theory

Abstract

By adapting arguments of Annala-Hoyois-Iwasa in the log setting, we prove Poincaré duality for smooth projective morphisms in logarithmic motivic homotopy theory. As an application, we show that the crystalline cohomology of a log compactification is independent of the choice.
Paper Structure (11 sections, 38 theorems, 146 equations)

This paper contains 11 sections, 38 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Saturated logarithmic motivic homotopy theory
  3. Thom space
  4. Log Gysin sequence
  5. Log Gysin morphisms are compatible with compositions
  6. A few commutative diagrams
  7. Evaluation morphism
  8. Poincaré duality
  9. Poincaré duality for blow-up squares
  10. w1-modules
  11. Failure of the localization property

Key Result

Theorem A

Let $f\colon X\to S$ be a projective morphism of schemes. Then

Theorems & Definitions (78)

  • Theorem A: See Theorem \ref{['local.2']}
  • Theorem B: See Theorem \ref{['local.3']}
  • Theorem C: See Theorem \ref{['coh.8']}
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • ...and 68 more