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}.Comment: 81 pages. Final version, to appear in the Actes of a 2016 conference in the Tata Institute. Thanks a lot goes to the referee for his enormous work (more than 100 comments) which was of great help. Among these corrections, he indicated to me a sign mistake in formula (3.2.14.a) which was very hard to detec