Galois Cohomology of Reductive Groups over Global Fields

We give closed formulas for the abelian Galois cohomology groups H ab 1 ( F , G ) H^1_{\operatorname {ab}}(F,G) and H ab 2 ( F , G ) H^2_{\operatorname {ab}}(F,G) of a connected reductive group G G over a global field F F in terms of the algebraic fundamental group π 1 ( G ) \pi _1(G) introduced earlier by one of us (M.B.). We further give closed formulas for the effects of restriction, corestriction, and localization, in terms of these formulas. Building on this, we give formulas, suitable for computer computations, for the first nonabelian Galois cohomology set H 1 ( F , G ) H^1(F,G) of G G and for the second Galois cohomology group H 2 ( F , T ) H^2(F,T) of an F F -torus T T . As a preparation for the derivation of our formulas, we review the interpretation of Tate cohomology of a finite group in terms of the stable derived category of Z [ Γ ] {\mathbb Z}[\Gamma ] -modules due to Buchweitz, and relate it to the explicit definition via cochains due to Kottwitz–Shelstad. We use this to construct the Tate–Nakayama isomorphisms for bounded complexes of tori over local and global fields