I want to do a series on the basics of Brauer groups since they came up in the past few posts. Since I haven’t really talked about Galois cohomology anywhere, we’ll take a slightly nonstandard approach and view everything “geometrically” in terms of étale cohomology. Everything should be equivalent to the Galois cohomology approach, but this way will allow us to use the theory that is already developed elsewhere on the blog.
I apologize in advance for the sporadic nature of this post. I just need to get a few random things out there before really starting the series. There will be one or two posts on the Brauer group of a “point” which will just mean the usual Brauer group of a field (to be defined shortly). Then we’ll move on to the Brauer group of a curve, and maybe if I still feel like continuing the series of a surface.
Let be a field and a fixed separable closure. We will define . This isn’t the usual definition and is often called the cohomological Brauer group. The usual definition is as follows. Let be a commutative, local, (unital) ring. An algebra over is called an Azumaya algebra if it is a free of finite rank -module and sending to is an isomorphism.
Define an equivalence relation on the collection of Azumaya algebras over by saying and are similar if for some and . The set of Azumaya algebras over modulo similarity form a group with multiplication given by tensor product. This is called the Brauer group of denoted . Often times, when an author is being careful to distinguish, the cohomological Brauer group will be denoted with a prime: . It turns out that there is always an injection .
One way to see this is that on the étale site of , the sequence of sheaves is exact. It is a little tedious to check, but using a Čech cocycle argument (caution: a priori the cohomology “groups” are merely pointed sets) one can check that the injection from the associated long exact sequence is the desired injection.
If we make the extra assumption that has dimension or , then the natural map is an isomorphism. I’ll probably regret this later, but I’ll only prove the case of dimension , since the point is to get to facts about Brauer groups of fields. If has dimension , then it is a local Artin ring and hence Henselian.
One standard lemma to prove is that for local rings a cohomological Brauer class comes from an Azumaya algebra if and only if there is a finite étale surjective map such that pulls back to in . The easy direction is that if it comes from an Azumaya algebra, then any maximal étale subalgebra splits it (becomes the zero class after tensoring), so that is our finite étale surjective map. The other direction is harder.
Going back to the proof, since is Henselian, given any class a standard Čech cocycle argument shows that there is an étale covering such that . Choosing any we have a finite étale surjection that kills the class and hence it lifts by the previous lemma.
It is a major open question to find conditions to make surjective, so don’t jump to the conclusion that we only did the easy case, but it is always true. Now that we have that the Brauer group is the cohomological Brauer group we can convert the computation of for a Henselian local ring to a cohomological computation using the specialization map (pulling back to the closed point) where .