Brauer Group

Abstract

This chapter continues the study of finite-dimensional associative division algebras over a field F, with particular attention to those that are simple and have center F. Section 5 is a selfcontained digression on cohomology of groups that is preparation for an application in Section 6 and for a general treatment of homological algebra in Chapter IV.

Section 1 introduces the Brauer group of F and the relative Brauer group of K/F, K being any finite extension field. The Brauer group B(F) is the abelian group of equivalence classes of finite-dimensional central simple algebras over F under a relation called Brauer equivalence. The inclusion F ⊆ K induces a group homomorphism B(F) → B(K), and the relative Brauer group B(K/F) is the kernel of this homomorphism. The members of the kernel are those classes such that the tensor product with K of any member of the class is isomorphic to some full matrix algebra M_n(K); such a class always has a representative A with dim_f A = (dim_f K)². One proves that B(F) is the union of all B(K/F) as K ranges over all finite Galois extensions of F.

Sections 2–3 establish a group isomorphism B(K/F)≌ H²(Gal(K/F), K^x) when K is a finite Galois extension of F. With these hypotheses on K and F, Section 2 introduces data called a factor set for each member of B(K/F). The data depend on some choices, and the effect of making different choices is to multiply the factor set by a “trivial factor set.” Passage to factor sets thereby yields a function from B(K/F) to the cohomology group H²(Gal(K/F), K^x). Section 3 shows how to construct a concrete central simple algebra over F from a factor set, and this construction is used to show that the function from B(K/F) to H²(Gal(K/F), K^x) constructed in Section 2 is one-one onto. An additional argument shows that this function in fact is a group isomorphism.

Section 4 proves under the same hypotheses that H¹(Gal(K/F), K^x) = 0, and a corollary makes this result concrete when the Galois group is cyclic. This result and the corollary are known as Hilbert’s Theorem 90.

Section 5 is a self-contained digression on the cohomology of groups. If G is a group and ZG is its integral group ring, a standard resolution of Z by free ZG modules is constructed in the category of all unital left ZG modules. This has the property that if M is an abelian group on which G acts by automorphisms, then the groups Hⁿ(G, M) result from applying the functor Hom_ZG(., M) to the members of this resolution, dropping the term Hom_ZG(Z, M), and taking the cohomology of the resulting complex. Section 5 goes on to show that the groups Hⁿ(G, M) arise whenever this construction is applied to any free resolution of Z, not necessarily the standard one. This section serves as a prerequisite for Section 6 and as motivational background for Chapter IV.

Section 6 applies the result of Section 5 in the case thatG is finite cyclic, producing a nonstandard free resolution of Z in this case. From this alternative free resolution, one obtains a rather explicit formula for H²(G, M) whenever G is finite cyclic. Application to the case that G is the Galois group Gal(K/F) for a finite Galois extension gives the explicit formula B(K/F)≌F^x/NK/F (K^x) for the relative Brauer group when the Galois group is cyclic.