Intro to Brauer Groups

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 {K} be a field and {K^s} a fixed separable closure. We will define {Br(K)=H^2_{et}(Spec(K), \mathbb{G}_m)=H^2(Gal(K^s/K), (K^s)^\times)}. This isn’t the usual definition and is often called the cohomological Brauer group. The usual definition is as follows. Let {R} be a commutative, local, (unital) ring. An algebra {A} over {R} is called an Azumaya algebra if it is a free of finite rank {R}-module and {A\otimes_R A^{op}\rightarrow End_{R-mod}(A)} sending {a\otimes a'} to {(x\mapsto axa')} is an isomorphism.

Define an equivalence relation on the collection of Azumaya algebras over {R} by saying {A} and {A'} are similar if {A\otimes_R M_n(R)\simeq A'\otimes_R M_{n'}(R)} for some {n} and {n'}. The set of Azumaya algebras over {R} modulo similarity form a group with multiplication given by tensor product. This is called the Brauer group of {R} denoted {Br(R)}. Often times, when an author is being careful to distinguish, the cohomological Brauer group will be denoted with a prime: {Br'(R)}. It turns out that there is always an injection {Br(R)\hookrightarrow Br'(R)}.

One way to see this is that on the étale site of {Spec(R)}, the sequence of sheaves {1\rightarrow \mathbb{G}_m\rightarrow GL_n\rightarrow PGL_n\rightarrow 1} 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 {H^1(Spec(R), PGL_n)/H^1(Spec(R), GL_n)\hookrightarrow Br'(R)} is the desired injection.

If we make the extra assumption that {R} has dimension {0} or {1}, then the natural map {Br(R)\rightarrow Br'(R)} is an isomorphism. I’ll probably regret this later, but I’ll only prove the case of dimension {0}, since the point is to get to facts about Brauer groups of fields. If {R} has dimension {0}, then it is a local Artin ring and hence Henselian.

One standard lemma to prove is that for local rings a cohomological Brauer class {\gamma\in Br'(R)} comes from an Azumaya algebra if and only if there is a finite étale surjective map {Y\rightarrow Spec(R)} such that {\gamma} pulls back to {0} in {Br'(Y)}. 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 {R} is Henselian, given any class {\gamma\in H^2(Spec(R), \mathbb{G}_m)} a standard Čech cocycle argument shows that there is an étale covering {(U_i\rightarrow Spec(R))} such that {\gamma|_{U_i}=0}. Choosing any {U_i\rightarrow Spec(R)} 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 {Br(X)\rightarrow Br'(X)} 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 {Br(R)} for a Henselian local ring to a cohomological computation using the specialization map (pulling back to the closed point) {Br(R)\rightarrow Br(k)} where {k=R/m}.