Mayer--Vietoris sequences for complexes of tori
Nguyen Manh Linh
TL;DR
This work extends patching techniques from linear algebraic groups to 2-term complexes of $F$-tori by developing crossed-module and bitorsor machinery, then establishing a six-term MV sequence for crossed modules and a nine-term MV sequence for short complexes of tori via (co-)flasque resolutions. The results hold without global domination assumptions on Galois cohomology and yield powerful applications: patching for nonabelian $\mathrm{H}^2$ of reductive groups with smooth centers, a weak local–global principle for simply connected semisimple groups, and a local–global principle for indices of central simple algebras. The approach connects $R$-equivalence on hypercohomology with flasque resolutions, enabling explicit identifications of Tate–Shafarevich-type obstructions and broadening the scope of HHK-type patching results to nonabelian cohomology settings.
Abstract
In the patching setting, given a factorization inverse system of fields over which patching for finite-dimensional vector spaces holds, together with a crossed module over the inverse limit field, the corresponding six-term Mayer--Vietoris sequence is constructed, generalizing the classical result of Harbater--Hartmann--Krashen for linear algebraic groups. When the crossed module is a two-term complex of tori, the above sequence is extended into a nine-term exact sequence, notably without any assumption on global domination of Galois cohomology of the inverse system. To this end, we develop a theory of (co-)flasque resolutions for short complexes of tori which is built on the work of Colliot-Thélène and Sansuc. As an application, we show that patching holds for nonabelian second Galois cohomology of reductive groups with smooth centers. We then obtain a weak local--global principle for this cohomology set in the simply connected semisimple case. We also rediscover a well-known local--global principle for indices of central simple algebras.