Th\'eor\`emes de dualit\'e pour les complexes de tores

TL;DR

This paper establishes local and global duality theorems for complexes of tori over number fields, generalizing existing duality results and impacting the understanding of arithmetic properties of algebraic groups.

Contribution

It proves a Poitou-Tate exact sequence for complexes of tori, extending duality theorems to more general algebraic structures over number fields.

Findings

Established duality theorems for complexes of tori.

Proved a Poitou-Tate exact sequence for these complexes.

Connected duality results to arithmetic properties of algebraic groups.

Abstract

We consider a complex of tori of length 2 defined over a number field k. We establish here some local and global duality theorems for the (\'etale or Galois) hypercohomology of such a complex. We prove the existence of a Poitou-Tate exact sequence for such a complex, which generalizes the Poitou-Tate exact sequences for finite Galois modules and tori. In particular, we obtain a Poitou-Tate exact sequence for k-groups of multiplicative type. The general results proven here lie at the root of recent results about the defect of strong approximation in connected linear algebraic groups and about some arithmetic duality theorems for the (non-abelian) Galois cohomology of such groups.