In this post we compute the group where is a number field.
A reminder on some definitions
The Brauer group of a scheme has several equivalent definitions. The first, the one most similar to the case one usually encounters (the Brauer groups of fields), involves so-called Azumaya algebras. Namely, let us call a quasicoherent sheaf of -algebras an Azumaya algebra of rank if there is a covering of in the étale topology such that .
As an example, if , where is a field, one can show that a -algebra satisfying the property that must also satisfy the property for some finite Galois extension . In particular, the Azumaya algebras on are just the central simple algebras on .
We shall call two Azumaya algebras equivalent if they are Morita equivalent. Specifically, we say is equivalent to if there are vector bundles on such that
as sheaves of algebras. One can show that the set of Azumaya algebras, up to equivalence, is a group under tensor product. We denote this group as .
Again, as an example, we see that if then the above becomes equivalent to the statement that two central simple algebras are equivalent if, as -algebras,
for some . Thus, .
Finally, if , we will often times denote as . This will cause no confusion in the classical case of fields, as mentioned in the last paragraph.
Some useful tools
Brauer group equals cohomological Brauer group (for affine schemes)
Before we go galavanting into the computation of we recall some important facts which will make the computation possible.
First, we have the following theorem:
Theorem: Let . Then, there is an isomorphism:
I am not positive to who the attribution of this theorem should go. It is certainly proven, in greater generality than this, in Gabber’s thesis (the results of which, as well as a generalization of, can be found here). While this theorem is difficult to prove in general, one can fairly easily describe the map which ends up being the isomorphism.
Note first that since rank Azumaya algebras on are étale locally , whose automorphism group is , we get a bijection (admittedly with some detail checking)
Thus, we get a bijection
One can show that this is even an isomorphism of groups when one gives the right hand side the group structure coming from the natural map
which endows the right hand side (together with the cup-product) with a group structure.
Now, the map is just the natural map one obtains from the above equivalence and the natural maps
one obtains from short exact sequence of sheaves
which is still exact on the étale site.
A little class field theory
The key result for this computation will be the following ‘fundamental exact sequence of (global) class field theory’
Theorem: Let be a global field. Then, the following sequence is short exact
where the first map is , ranges over the places of , and the last map is the so-called ‘invariant’ map. To be more specific, if one identifies each with (which one can do), the invariant map is just summing the coordinates.
Note that this is indeed a deep theorem. Contained in its fold (specifically the injectivity of the first map) is the Brauer-Hasse-Arf theorem that a central simple algebra is split (i.e. is a matrix algebra over ) if and only if it is split locally (i.e. is split for all ). Applying this to the case of quaternion algebras over yields the famous Hasse pricinple for quadratic forms: that a quadratic form has a solution in if and only if it has a solution in for all .
Now, using the fact that
we can non-canonically write the above sequence as
where is the number of real places of .
A short exact sequence
The last important tool for us will be the existence of a certain short exact sequence of sheaves on . This is a sequence that anyone familiar with the computation of the étale cohomology of curves (cf. SGA 4.5) will likely remember.
Theorem: Let be an integral, separated, locally factorial scheme with function field . Let denote the inclusion of its generic point. Then, the following sequence is exact:
where the first map is the obvious one, and the second map is the ‘divisor’ map.
Here is the sheaf
is the ‘sheaf of Weil divisors’ on .
Proving this theorem is not difficult, and is a good exercise in working in the étale topology.
So, we can finally get to the computation of .
So, we begin, as we usually do, with the relevant exact sequence. Thus, if , and everything else is in the last section, then we have the following exact sequence
Gussying this up a little bit gives:
Now, some of these terms are easy to compute. For example,
where the first isomorphism is because cohomology commutes with direct sums in this case, and the second since is a finite map.
But, using the fact that étale cohomology on the spectrum of a field is just Galois cohomology, and under this correspondence constant sheaves go to trivial modules we see that
The computation of the second cohomology of succumbs to similar methods. Namely, the same reasoning shows that
where the cohomology on the right hand side is continous cohomology with a trivial module.
The last relevant calculation we need to do is the computation of . To do this we appeal to the Leray spectral sequence which gives us
But, we claim that for . To do this, it suffices to check that it vanishes on geometric points centered at closed points. So, let be a geometric point of . Then, by standard theory we know that
where is the fraction field of strict henselization of at . One can verify that if hits the point of then
So, we need to prove that the Galois cohomology of with respect to is zero in positive degree. This is non-trivial. For this is Hilbert’s theorem 90. For we’re asking about the Brauer group of the fraction field of a strictly Henselian ring–this is zero (cf. Milne’s book). Then, in general, one can use the difficult fact that the cohomological dimension of is , together with the Kummer sequence to obtain the result.
Regardless, we see that the Leray spectral sequence degenerates on the first page and so we have that
Thus, we have proven that there is an exact sequence
This seems relatively unhelpful since we don’t have a handle on this second map. The key observation though is that
and that the diagram
commutes. If this seems unbelievable, it’s because you’ve forgotten that this is precisely how one proves that (see, for example, Serre’s relevant article in Cassels and Frolich)!
Thus, we see that we can rewrite our sequence as
But, combining this with the fundamental sequence of class field theory we obtain the following short exact sequence
where is zero or depending on if has a real place or not. But, this sequence splits (it’s a sequence of -spaces!) and so we can finally conclude that