Abelian Galois Cohomology of Reductive Groups
نویسندگان
چکیده
In this paper we introduce a new functor H ab(K,G) from the category of connected reductive groups over a field K of characteristic 0 to abelian groups. We call H ab(K,G) the first abelian Galois cohomology group of G. The abelian group H ab(K,G) is related to the pointed setH (K,G) by the abelianization map ab:H(K,G)→ H ab(K,G). When K is a local field or a number field, the map ab 1 is surjective. We use the map ab to give a functorial, almost explicit description of the set H(K,G) when K is a number field. We describe the contents in more detail. For a reductive group G over a field K of characteristic 0, let G denote the derived group of G (it is semisimple) and let G denote the universal covering group of G (it is simply connected). Following Deligne we consider the composition
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