Cohomology and Galois theory. I. Normality of algebras and Teichmüller’s cocycle

An action of the automorphism group of a commutative ring on its Brauer group

“…It follows that Aut (22: S) acts on H n (G, U(S)) and H n (G, Pic (S)) for all n^O. This action is given in [11] (G, S) given by Φn(s (x) (x) sj(r fc , ..., τ Λ ) = s&Mτ&M -zv τ ft (s ft+1 ) for s, e S and τ t e G induce a homomorphism Ffe): 2^(S* +1 ) -*F{K\G, S)) = K*(G, F(S)) which induces a morphism of complexes for any additive functor F. This morphism of complexes induces homomorphisms <ψ n \ H n (S/R,…”

“…Let L be a finite Galois field extension of K with finite Galois group G and let Aut (K: L) be the group of automorphisms of K which can be extended to L. In [11] S. Eilenberg and S. MacLane gave an action of Aut (K: L) on H n (G, L*) for n ^> 0. This action corresponds under the natural identification between B{LjK) and H 2 …”

Inertial subalgebras of algebras possessing finite automorphism groups

“…If, moreover, A is separable over C, then A will be said to be a normal central separable /^-algebra with center C and group G. (We note that this terminology is somewhat at variance with that of Eilenberg and MacLane (1948), Pareigis (1964) and Childs (1964). In these papers G is a set, not necessarily a group, of i?-automorphisms of A which restricts faithfully to a group of ^-automorphisms of C with respect to which C is a Galois extension of R.) Let C be a (G, 7?…”

The exact sequence of low degree and normal algebras

“…The terms arising from the homology of the total complex are not F n~"1 H n (tot) i the (n-l)th filtration group of H n , for n>2, but map onto it. As an application we embed the seven term Galois cohomology sequence of [l, 5.5] into an infinite sequence, and sketch a map from normal Azumaya algebras into the eighth term which extends the Teichmüller cocycle map of [3]. The nth group C n (tot) of the total complex (n^O) is the group…”

“…We now sketch a generalization of the Teichmuller cocycle map for normal central simple algebras described by Eilenberg and MacLane [3]. For Amitsur cohomology and other unexplained notation see [2].…”