18 - A Survey of some further Results

Summary

There are many aspects of Brauer groups which we cannot discuss here for various reasons.

First, the relationship with Galois Cohomology (cf. §§12/13. where we established an isomorphism Br(L/K) ≃ H²(Gal(L/K), L^*) ≕ H²(L/K) for Galois extensions L/K) leads to further results if one makes use of general Cohomology Theory: then some of our results appear as special cases of rather general constructions (such as Theorem 4 in §9. and the exact sequence (4) in §13. which are both easy consequences of the Hochschild–Serre spectral sequence). A good reference for this point of view is A. Babakhanian [1972] or E. Weiss [1969].

Second, the relationship with cohomology (see above) may be generalized in the following way: one may establish an isomorphism Br(L/K) ≃ H²(L/K) even when L/K is not Galois but separable. Then, of course, H²(L/K) has to be given a new meaning: it is no more a Galois Cohomology group H²(Gal(L/K), L^*) but an Adamson Cohomology group. Here we refer the reader to the original paper I. T. Adamson [1954].

Third, by introducing even more general cohomology groups – the Amitsur Cohomology groups – one can obtain for instance all our results on Br(K) without making use of Köthe's Theorem (as we do frequently).