# Orientation theory in arithmetic geometry

Abstract : This work is devoted to study orientation theory in arithmetic geometric within the motivic homotopy theory of Morel and Voevodsky. The main tool is a formulation of the absolute purity property for an \emph{arithmetic cohomology theory}, either represented by a cartesian section of the stable homotopy category or satisfying suitable axioms. We give many examples, formulate conjectures and prove a useful property of analytical invariance. Within this axiomatic, we thoroughly develop the theory of characteristic and fundamental classes, Gysin and residue morphisms. This is used to prove Riemann-Roch formulas, in Grothendieck style for arbitrary natural transformations of cohomologies, and a new one for residue morphisms. They are applied to rational motivic cohomology and \'etale rational $\ell$-adic cohomology, as expected by Grothendieck in \cite[XIV, 6.1]{SGA6}.
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https://hal.archives-ouvertes.fr/hal-02357234
Contributor : Frédéric Déglise <>
Submitted on : Saturday, November 9, 2019 - 5:02:27 PM
Last modification on : Sunday, November 17, 2019 - 11:22:34 AM

### Identifiers

• HAL Id : hal-02357234, version 1
• ARXIV : 1111.4203

### Citation

Frédéric Déglise. Orientation theory in arithmetic geometry. K-Theory - Proceedings of the International Colloquium, Mumbai, 2016, 2018. ⟨hal-02357234⟩

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