Orientation theory in arithmetic geometry

TL;DR

This paper develops orientation theory within arithmetic geometry using motivic homotopy theory, establishing foundational properties, conjectures, and formulas like Riemann-Roch for various cohomology theories.

Contribution

It introduces an axiomatic framework for orientation in arithmetic geometry, proving new Riemann-Roch formulas and analyzing characteristic classes and residue morphisms.

Findings

Proves a form of absolute purity for arithmetic cohomology theories.

Develops a comprehensive theory of characteristic and fundamental classes.

Establishes Riemann-Roch formulas for rational motivic and étale cohomology.

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}.