Groupes algébriques sur un corps local. III: Compléments et applications à la cohomologie galoisienne. (Algebraic groups over a local field. III: Comments and applications to Galois cohomology). (French) Zbl 0657.20040

This is a continuation of the authors’ previous papers [Publ. Math., Inst. Hautes Etud. Sci. 41, 5-251 (1972; Zbl 0254.14017); 60, 1-194 (1984; Zbl 0597.14041)]. Let K be a complete field with respect to a discrete valuation which has the perfect residue field k. The reductive algebraic groups G defined over K can be regarded as “infinite dimensional objects over k” and the theory of the groups has some striking analogies with the classical theory of reductive groups over an arbitrary field L which was initiated by N. Iwahori and H. Matsumoto. Let \(DG^ 0\) be the derived group of the connected component of the identity \(G^ 0\) of G. The group G defined over K is said to be residually split over K if the rank of \(DG^ 0\) over K is equal to that over the maximal unramified extension \(\tilde K\) of K. Also G is said to be residually quasi-split over K if it has an Iwahori subgroup which is stable under the Galois group \(Gal(\tilde K/K)\). These are the analogues of the notion of the groups split or quasi-split over L but here the role of the separable closure of L is replaced by the extension \(\tilde K\) of K. The authors discuss the problem of the existence and uniqueness of a residually split group over K which is \(\tilde K-\)isomorphic to the given group defined over K. This is not so simple and there are some differences between the case of absolutely almost simple and the general case. The authors give applications of the theory to study the Galois cohomology of the groups. In particular, it is shown that if the residue field k is of cohomological dimension \(\leq 1\), the group G defined over K is residually quasi-split and further if G is connected, simply connected, then \(H^ 1(G)=0\). In this case, all anisotropic groups over K can be determined explicitly. This generalizes and simplifies the proof of results obtained by M. Kneser when K is a locally compact field of characteristic 0.
Reviewer: E.Abe


20G25 Linear algebraic groups over local fields and their integers
20G10 Cohomology theory for linear algebraic groups
14L35 Classical groups (algebro-geometric aspects)