Galois cohomology of reductive groups over global fields

TL;DR

This work develops a comprehensive, computable description of abelian and nonabelian Galois cohomology for reductive groups over global fields. It extends Tate cohomology to bounded complexes of Γ-modules via the stable derived category, and proves Tate–Nakayama isomorphisms locally and globally for complexes of tori. These tools yield explicit, finite, Cartesian descriptions of $H^1(F,G)$ and $H^2(F,T)$ in terms of the algebraic fundamental group $\,\pi_1(G)$ and real place data, with full functorial compatibility under restriction, localization, and connecting maps. The results enable practical computations of Galois cohomology and Tate–Shafarevich kernels, linking local-global structures to stable crossed modules and providing a robust framework for explicit arithmetic of reductive groups. The framework unifies and extends classical descriptions (e.g., Kottwitz, Tate–Nakayama) and offers algorithmic paths for calculating cohomology and Shafarevich groups in both local and global settings.

Abstract

We give closed formulas for the abelian Galois cohomology groups H^1_{ab}(F,G) and H^2_{ab}(F,G) of a connected reductive group G over a global field F in terms of the algebraic fundamental group π_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 and the analogous known formulas in the case of local fields. Building on this, we give formulas, suitable for computer computations, for the first nonabelian Galois cohomology set H^1(F,G) of G and for the second Galois cohomology group H^2(F,T) of an F-torus 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]-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, whose specialization to complexes of length 2 is then applied to obtain the desired formulas.

Theorems & Definitions (215)

• Theorem 1.5.1
• Remark 1.5.2
• Remark 1.5.3
• Remark 1.5.4
• Remark 1.5.5
• Theorem 1.6.1: Theorem \ref{['t:explicit']}
• Theorem 1.6.2: Corollary \ref{['c:not-direct']}
• Theorem 1.7.1: Theorem \ref{['thm:tnglob']}
• Proposition 2.5.1
• proof