Abelian Galois cohomology of quasi-connected reductive groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914556324872192 |
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| author | Borovoi, Mikhail Kang, Taeyeoup |
| author_facet | Borovoi, Mikhail Kang, Taeyeoup |
| contents | In 1999 Labesse introduced quasi-connected reductive groups and investigated their abelian Galois cohomology over local and global fields of characteristic 0. We (1) generalize some of the constructions of Labesse from quasi-connected reductive groups to arbitrary reductive groups, not necessarily connected or quasi-connected; (2) generalize results of Labesse on the abelian Galois cohomology of quasi-connected reductive groups to the case of local and global fields of arbitrary characteristic; and (3) investigate the functoriality properties of the abelian Galois cohomology. In particular, we introduce the notion of a principal homomorphism of quasi-connected reductive groups, and show that if G is a quasi-connected reductive group over a local or global field k of *positive* characteristic, then the first Galois cohomology set H^1(k,G) has a canonical structure of abelian group, which is functorial with respect to *principal* homomorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22059 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Abelian Galois cohomology of quasi-connected reductive groups Borovoi, Mikhail Kang, Taeyeoup Representation Theory Algebraic Geometry Group Theory 11E72, 14L15, 20G10, 20G15, 20G20, 20G25, 20G30, 20G35 In 1999 Labesse introduced quasi-connected reductive groups and investigated their abelian Galois cohomology over local and global fields of characteristic 0. We (1) generalize some of the constructions of Labesse from quasi-connected reductive groups to arbitrary reductive groups, not necessarily connected or quasi-connected; (2) generalize results of Labesse on the abelian Galois cohomology of quasi-connected reductive groups to the case of local and global fields of arbitrary characteristic; and (3) investigate the functoriality properties of the abelian Galois cohomology. In particular, we introduce the notion of a principal homomorphism of quasi-connected reductive groups, and show that if G is a quasi-connected reductive group over a local or global field k of *positive* characteristic, then the first Galois cohomology set H^1(k,G) has a canonical structure of abelian group, which is functorial with respect to *principal* homomorphisms. |
| title | Abelian Galois cohomology of quasi-connected reductive groups |
| topic | Representation Theory Algebraic Geometry Group Theory 11E72, 14L15, 20G10, 20G15, 20G20, 20G25, 20G30, 20G35 |
| url | https://arxiv.org/abs/2603.22059 |