Chapter III. 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 self-contained 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 $\mathcal{B}(F)$ is the abelian group of equivalence classes of finite-dimensional central simple algebras over $F$ under a relation called Brauer equivalence. The inclusion $F\subseteq K$ induces a group homomorphism $\mathcal{B}(F)\to\mathcal{B}(K)$, and the relative Brauer group $\mathcal{B}(K/F)$ 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 $\dim_FA=(\dim_FK)^2$. One proves that $\mathcal{B}(F)$ is the union of all $\mathcal{B}(K/F)$ as $K$ ranges over all finite Galois extensions of $F$.

Sections 2–3 establish a group isomorphism $\mathcal{B}(K/F)\cong H^2(\mathrm{Gal}(K/F),K^{\times})$ 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 $\mathcal{B}(K/F)$. 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 $\mathcal{B}(K/F)$ to the cohomology group $H^2(\mathrm{Gal}(K/F),K^{\times})$. 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 $\mathcal{B}(K/F)$ to $H^2(\mathrm{Gal}(K/F),K^{\times})$ 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 $H^1(\mathrm{Gal}(K/F),K^{\times})=0$, 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 $\mathbb{Z} G$ is its integral group ring, a standard resolution of $\mathbb{Z}$ by free $\mathbb{Z} G$ modules is constructed in the category of all unital left $\mathbb{Z} G$ modules. This has the property that if $M$ is an abelian group on which $G$ acts by automorphisms, then the groups $H^n(G,M)$ result from applying the functor $\mathrm{Hom}_{\mathbb{Z} G}(\,\cdot\,,M)$ to the members of this resolution, dropping the term $\mathrm{Hom}_{\mathbb{Z} G}(\mathbb{Z},M)$, and taking the cohomology of the resulting complex. Section 5 goes on to show that the groups $H^n(G,M)$ arise whenever this construction is applied to any free resolution of $\mathbb{Z}$, 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 that $G$ is finite cyclic, producing a nonstandard free resolution of $\mathbb{Z}$ in this case. From this alternative free resolution, one obtains a rather explicit formula for $H^2(G,M)$ whenever $G$ is finite cyclic. Application to the case that $G$ is the Galois group $\mathrm{Gal}(K/F)$ for a finite Galois extension gives the explicit formula $\mathcal{B}(K/F)\cong F^{\times}\big/N_{K/F}(K^{\times})$ for the relative Brauer group when the Galois group is cyclic.