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arXiv · 1111.4203

Orientation theory in arithmetic geometry

Frédéric Déglise
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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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Frédéric Déglise. 2011-11-17. Orientation theory in arithmetic geometry. https://arxiv.org/abs/1111.4203

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