Galois cohomology of reductive groups over global fields

2023/03/07 by Borovoi, Mikhail, Kaletha, Tasho, Hinich, Vladimir · 1 citation
#11E72 #20G10 #20G25 #20G30 #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2303.04120

Abstract

We give closed formulas for the abelian Galois cohomology groups H¹_ab(F,G) and H²_ab(F,G) of a connected reductive group G over a global field F in terms of the algebraic fundamental group π₁(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 H¹(F,G) of G and for the second Galois cohomology group H²(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.

Cited by

• Cup product of inhomogeneous Tate cochains, and application to tori over local fields that split over cyclic extensions

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