Galois cohomology of reductive groups over global fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912363816419328 |
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| author | Borovoi, Mikhail Kaletha, Tasho Hinich, Vladimir |
| author_facet | Borovoi, Mikhail Kaletha, Tasho Hinich, Vladimir |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_04120 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Galois cohomology of reductive groups over global fields Borovoi, Mikhail Kaletha, Tasho Hinich, Vladimir Number Theory Algebraic Geometry Group Theory Representation Theory 11E72, 20G10, 20G25, 20G30 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. |
| title | Galois cohomology of reductive groups over global fields |
| topic | Number Theory Algebraic Geometry Group Theory Representation Theory 11E72, 20G10, 20G25, 20G30 |
| url | https://arxiv.org/abs/2303.04120 |