Patching and the nine-term Mayer-Vietoris sequence for complexes of tori

We present the patching method, a machinery developed by Harbater–Hartmann–Krashen and various other authors, dedicated to the study of arithmetics of linear algebraic groups over function fields of curves over complete discretely valued field such as ℚₚ(T). Then, we present a new result in this direction, which gives a nine 9-term exact sequence for Galois cohomology of 2-term complexes of tori in the patching setting. This relies on the notion of (co-)flasque resolutions of such complexes, generalizing the previous work by Colliot-Thélène–Sansuc. As applications, we show that patching holds for nonabelian second Galois cohomology of reductive groups with a smooth center, as well as a weak local–global principle for this cohomology set. We also rediscover a local–global principle for indices of central simple algebras.