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Abelian Galois cohomology of reductive groups. (English) Zbl 0918.20037

Mem. Am. Math. Soc. 626, 50 p. (1998).
Denote by \(G\) a reductive algebraic group over a field \(K\) of characteristic 0 and by \(T\) a maximal torus of \(G\). In this paper new functors \[ H^i_{ab}:=\mathbb{H}^i(K,T^{(sc)}\to T)\quad(i\geq-1) \] are defined as the abelian Galois cohomology groups, where \(\mathbb{H}^i(K,T^{(sc)}\to T)\) is the Galois hypercohomology of the complex \(T^{sc}(\overline K)\to T(\overline K)\).
This definition allows the author to generalize the abelian cohomology to arbitrary reductive groups (and indeed to arbitrary algebraic groups, modulo the unipotent radical) over a field of characteristic zero; and to obtain even in this case some result proved by Sansuc for semisimple groups, by Kottwitz, Langlands for local fields and by Kneser, Harder and Chernousov for number fields. The definition of two abelianization maps \[ ab^0\colon H^0(K,G)\to H^0_{ab}(K,G)\qquad ab^1\colon H^1(K,G)\to H^1_{ab}(K,G) \] relates the abelian cohomology to the usual one: for local non archimedean fields \(ab_0\) is a surjective homomorphism, and for number fields \(ab_1\) is surjective.

MSC:

20G10 Cohomology theory for linear algebraic groups
11E72 Galois cohomology of linear algebraic groups
14E20 Coverings in algebraic geometry
18G50 Nonabelian homological algebra (category-theoretic aspects)
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