2026/03/31 by Mikhail Borovoi, Taeyeoup Kang
Mathematics · #math.RT #math.AG #math.GR #msc:11E72 #msc:14L15 #msc:20G10 #msc:20G15 #msc:20G20 #msc:20G25 #msc:20G30 #msc:20G35
V.1: 42 pages, v.2: 44 pages, v.3: 51 pages. With a solution of Exercise 6.5.2(i) of Brian Conrad's text [Con14] (the existence of a universal covering of a semisimple group scheme over a non-empty base scheme). V.4 (final) 58 pages
arxiv created 2026/08/05 · arxiv updated 2026/08/06
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 H1(k,G) has a canonical structure of abelian group, which is functorial with respect to *principal* homomorphisms.