6533b82bfe1ef96bd128e760

RESEARCH PRODUCT

Orientation theory in arithmetic geometry

Frédéric Déglise

subject

Mathematics - Algebraic Geometryresiduescobordism14C40 14F42 14F20 19E20 19D45 19E15Mathematics::K-Theory and HomologyMathematics::Category Theory[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG][MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]Orientation theorymotivic homotopyMathematics::Algebraic TopologyRiemann-Roch formulas

description

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}.

yearjournalcountryeditionlanguage
2016-01-06
https://hal.science/hal-02357234