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Abelian Galois cohomology of quasi-connected reductive groups

Mikhail Borovoi, Taeyeoup Kang

Abstract

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.

Abelian Galois cohomology of quasi-connected reductive groups

Abstract

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 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.
Paper Structure (11 sections, 59 theorems, 243 equations)

This paper contains 11 sections, 59 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. A crossed module from a non-connected reductive group
  3. Crossed cohomology and the crossing map
  4. Quasi-connected reductive groups and principal homomorphisms
  5. A Picard crossed module from a quasi-connected reductive group
  6. A $t$-extension of a principal homomorphism
  7. Abelian Galois cohomology via the Picard braiding
  8. Galois cohomology over non-archimedean local fields
  9. Universal covering of a semisimple group scheme as a functor
  10. Coboundary for a Galois hyper-cocycle of a crossed module
  11. Group cohomology of a braided crossed module

Key Result

Theorem 1.1

Theorems & Definitions (138)

  • Theorem 1.1: Theorems \ref{['t:cr-surjective']} and \ref{['t:cr-bijective']}
  • Definition 1.2
  • Corollary 1.3
  • Example 1.4
  • Definition 1.5
  • Theorem 1.6: Corollary \ref{['c:principal-induces']}
  • Corollary 1.7: Theorem \ref{['t:functor']}
  • Theorem 1.8
  • Theorem 1.9: Theorem \ref{['t:t-ext-principal']}, not easy
  • Proposition 2.2
  • ...and 128 more