Contemporary Mathematics Volume 272, 2000 GALOIS COHOMOLOGY OF THE CLASSICAL GROUPS Eva Bayer-Fluckiger Introduction Galois cohomology sets of linear algebraic groups were first studied in the late 50's- early 60's. As pointed out in [18], for classical groups, these sets have classical interpretations. In particular, Springer's theorem [22] can be reformulated as an injectivity statement for Galois cohomology sets of orthogonal groups well-known classification results for quadratic forms over certain fields (such as finite fields, J:r-adic fields, ... ) correspond to vanishing of such sets. The language of Galois cohomology makes it possible to formulate analogous statements for other linear algebraic groups. In [18] and [20], Serre raises questions and conjectures in this spirit. The aim of this paper is to survey the results obtained in the case of the classical groups. 1. Definitions and notation Let k be a field of characteristic =f=. 2, let ks be a separable closure of k and let rk = Gal(ks/k). 1.1. Algebras with involution and norm-one-groups {cf. [9], [15]). Let A be a finite dimensional k-algebra. An involution a : A ---+ A is a k-linear antiau- tomorphism of A such that a 2 = id. Let (A, a) be an algebra with involution. The associated norm-one-group U A is the linear algebraic group over k defined by UA(E) ={a E A®E iaa(a) = 1} for every commutative k-algebra E. 1.2. Galois cohomology {cf. [20]). For any linear algebraic group U defined over k, set H 1 (k, U) = H 1 (rk, U(ks)). Recall that H 1 (k, U) is also the set of isomorphism classes of U-torsors (principal homogeneous spaces over U). 1.3. Cohomological dimension. Let k be a perfect field. We say that the cohomological dimension of k is ::: n, denoted by cd(k) ::: n, if Hi(rk, C) = 0 for every i nand for every finite rk-module C. @2000 American Mathematical Society http://dx.doi.org/10.1090/conm/272/04392
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