2009/06/18 by Cyril Demarche, Demarche, Cyril · 1 citation
Mathematics · #11R34 #12G05 #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.0906.3453
openalex publication_date 2009/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (étale 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.