  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}.
