Existence of Invariant Bases

Ellis Kolchin, Serge Lang
1960 Proceedings of the American Mathematical Society  
Let 7<" be a field, G a group of automorphisms of K, and M a vector space over K on which G acts in such a way that a(aD) = aa-crD for &EG, aEK, and DEM. The problem arises to find whether M has a basis consisting of invariant elements under G. In other words, letting K0 be the fixed field under G, and M0 the set of fixed elements of M under G so that Mo is a vector space over Ko, to find out whether M is isomorphic to the tensor product M « K ®K, Mo under the natural map. We shall see that
more » ... shall see that this is so if and only if a certain cocycle of G in the full linear group is trivial. In some applications, a rational structure is added to K and G, namely K is the function field of a principal homogeneous space over a group variety G. We shall show that the cocycle involved is then determined rationally. This leads us into a discussion of rational cocycles in §3, and of their comparison with the ordinary cocycles of Galois theory, i.e. where G is a finite Galois group. All cocycles involved with coefficients in the full linear group split, and in fact the Galois cohomology (in dimension 1) of the group variety of units in an algebra is trivial (Propositions 2 and 5).
doi:10.2307/2032732 fatcat:6mza6qbnqneg3h6asbd5juzikm