# A computation a day: the Brauer group of a number ring

In this post we compute the group $\mathrm{Br}(\mathcal{O}_K)$ where $K$ is a number field.

# A reminder on some definitions

The Brauer group of a scheme $X$ 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 $\mathcal{O}_X$-algebras $\mathcal{A}$ an Azumaya algebra of rank $n$ if there is a covering $\{U_i\}$ of $X$ in the étale topology such that $\mathcal{A}_{U_i}\cong \mathrm{Mat}_n(\mathcal{O}_{U_i})$.

As an example, if $X=\mathrm{Spec}(k)$, where $k$ is a field, one can show that a $k$-algebra $A$ satisfying the property that $A\otimes_k\overline{k}=\mathrm{Mat}_n(\overline{k})$ must also satisfy the property $A\otimes_k K\cong \mathrm{Mat}_n(K)$ for some finite Galois extension $K/k$. In particular, the Azumaya algebras on $\mathrm{Spec}(k)$ are just the central simple algebras on $k$.

We shall call two Azumaya algebras equivalent if they are Morita equivalent. Specifically, we say $\mathcal{A}$ is equivalent to $\mathcal{B}$ if there are vector bundles $\mathcal{E},\mathcal{E}'$ on $X$ such that

$\mathcal{A}\otimes_{\mathcal{O}_X}\mathrm{End}_{\mathcal{O}_X}(\mathcal{E})\cong \mathcal{B}\otimes_{\mathcal{O}_X}\mathrm{End}_{\mathcal{O}_X}(\mathcal{E}')$

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 $\mathrm{Br}(X)$.

Again, as an example, we see that if $X=\mathrm{Spec}(k)$ then the above becomes equivalent to the statement that two central simple algebras $A,B$ are equivalent if, as $k$-algebras,

$\mathrm{Mat}_n(A)\cong\mathrm{Mat}_m(B)$

for some $m,n$. Thus, $\mathrm{Br}(\mathrm{Spec}(k))=\mathrm{Br}(k)$.

Finally, if $X=\mathrm{Spec}(R)$, we will often times denote $\mathrm{Br}(X)$ as $\mathrm{Br}(R)$. 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 $\mathrm{Br}(\mathcal{O}_K)$ we recall some important facts which will make the computation possible.

First, we have the following theorem:

Theorem: Let $X=\mathrm{Spec}(R)$. Then, there is an isomorphism:

$\mathrm{Br}(R)\cong H^2(X_\mathrm{\acute{e}t},\mathbf{G}_m)$

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 $\mathrm{Br}(X)\to H^2(X_\mathrm{\acute{e}t},\mathbf{G}_m)$ which ends up being the isomorphism.

Note first that since rank $n$ Azumaya algebras on $X$ are étale locally $\mathrm{Mat}_n$, whose automorphism group is $\mathrm{PGL}_n$, we get a bijection (admittedly with some detail checking)

$\left\{\begin{matrix}\text{rank }n\text{ Azumaya}\\ \text{algebras on }X\end{matrix}\right\}\leftrightarrow H^1(X_{\mathrm{\acute{e}t}},\mathrm{PGL}_n)$

Thus, we get a bijection

$\mathrm{Br}(X)\xrightarrow{\approx}H^1(X_{\mathrm{\acute{e}t}},\mathrm{PGL}_\infty)=\varinjlim H^1(X_\mathrm{\acute{e}t},\mathrm{PGL}_n)$

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

$\mathrm{PGL}_n\otimes\mathrm{PGL}_m\to\mathrm{PGL}_{n+m}$

which endows the right hand side (together with the cup-product) with a group structure.

Now, the map $\mathrm{Br}(X)\to H^2(X_{\mathrm{\acute{e}t}},\mathbf{G}_m)$ is just the natural map one obtains from the above equivalence and the natural maps

$H^1(X_{\mathrm{\acute{e}t}},\mathrm{PGL}_n)\to H^2(X_{\mathrm{\acute{e}t}},\mathbf{G}_m)$

one obtains from short exact sequence of sheaves

$1\to \mathbf{G}_m\to\mathrm{GL}_n\to\mathrm{PGL}_n\to 1$

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 $K$ be a global field. Then, the following sequence is short exact

$\displaystyle 0\to\mathrm{Br}(K)\to\bigoplus_{v}\mathrm{Br}(K_v)\to\mathbb{Q}/\mathbb{Z}\to 0$

where the first map is $A\mapsto (A\otimes_K K_v)$, $v$ ranges over the places of $K$, and the last map is the so-called ‘invariant’ map. To be more specific, if one identifies each $\mathrm{Br}(K_v)$ with $\mathbb{Q}/\mathbb{Z}$ (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 $A/K$ is split (i.e. is a matrix algebra over $K$) if and only if it is split locally (i.e. $A\otimes_K K_v$ is split for all $v$). Applying this to the case of quaternion algebras over $K$ yields the famous Hasse pricinple for quadratic forms: that a quadratic form has a solution in $K$ if and only if it has a solution in $K_v$ for all $v$.

Now, using the fact that

$\mathrm{Br}(K_v)=\begin{cases}\mathbb{Q}/\mathbb{Z} & \mbox{if}\quad v\text{ is finite}\\ \mathbb{Z}/2\mathbb{Z} & \mbox{if}\quad v\text{ is real}\\ 0 & \mbox{if}\quad v\text{ is complex}\end{cases}$

we can non-canonically write the above sequence as

$\displaystyle 0\to\mathrm{Br}(K)\to \left(\bigoplus_{\mathfrak{p}\in\mathrm{Spec}(\mathcal{O}_K)}\mathbb{Q}/\mathbb{Z}\right)\oplus \left(\mathbb{Z}/2\mathbb{Z}\right)^r \to \mathbb{Q}/\mathbb{Z}\to 0$

where $r$ is the number of real places of $K$.

## A short exact sequence

The last important tool for us will be the existence of a certain short exact sequence of sheaves on $X_{\mathrm{\acute{e}t}}$. 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 $X$ be an integral, separated, locally factorial scheme with function field $K$. Let $\eta:\mathrm{Spec}(K)\to X$ denote the inclusion of its generic point. Then, the following sequence is exact:

$0\to \mathbf{G}_{m,X}\to \eta_\ast \mathbf{G}_{m,\mathrm{Spec}(K)}\to \mathrm{Div}_X\to 0$

where the first map is the obvious one, and the second map is the ‘divisor’ map.

Here $\mathrm{Div}_X$ is the sheaf

$\displaystyle \bigoplus_{\stackrel{x\in X}{\mathrm{codim}_X(x)=1}}(i_x)_\ast \underline{\mathbb{Z}}$

is the ‘sheaf of Weil divisors’ on $X$.

Proving this theorem is not difficult, and is a good exercise in working in the étale topology.

# Computation

So, we can finally get to the computation of $\mathrm{Br}(\mathcal{O}_K)=H^2\left(\left(\mathrm{Spec}(\mathcal{O}_K)\right)_{\mathrm{\acute{e}t}},\mathbf{G}_m\right)$.

So, we begin, as we usually do, with the relevant exact sequence. Thus, if $X=\mathrm{Spec}(\mathcal{O}_K)$, and everything else is in the last section, then we have the following exact sequence

$\displaystyle 0\to \mathcal{O}_K^\times\to K^\times\to \bigoplus_{\mathfrak{p}\in\mathrm{MaxSpec}(\mathcal{O}_K)}\mathbb{Z}\to H^1(X_\mathrm{\acute{e}t},\mathbf{G}_{m,X})\to H^1(X,\eta_\ast\mathbf{G}_{m,\mathrm{Spec}(K)})\to H^1(X_{\mathrm{\acute{e}t}},\mathrm{Div}_X)\to H^2(X_\mathrm{\acute{e}t},\mathbf{G}_{m,X})\to H^2(X_{\mathrm{\acute{e}t}},\eta_\ast\mathbf{G}_{m,\mathrm{Spec}(K)})\to H^2(X_{\mathrm{\acute{e}t}},\mathrm{Div}_X)$

Gussying this up a little bit gives:

$\displaystyle 0\to \mathcal{O}_K^\times\to K^\times\to \mathbb{Z}^{\mathrm{MaxSpec}(\mathcal{O}_K)}\to \mathrm{Cl}(K)\to H^1(X_{\mathrm{\acute{e}t}},\eta_\ast\mathbf{G}_{m,\mathrm{Spec}(K)})\to H^1(X_{\mathrm{\acute{e}t}},\mathrm{Div}_X)\to \mathrm{Br}(\mathcal{O}_K)\to H^2(X_{\mathrm{\acute{e}t}},\eta_\ast\mathbf{G}_{m,\mathrm{Spec}(K)})\to H^2(X,\mathrm{Div}_X)$

Now, some of these terms are easy to compute. For example,

\displaystyle \begin{aligned}H^1(X_{\mathrm{\acute{e}t}},\mathrm{Div}_X) &=\bigoplus_{\mathfrak{p}\in\mathrm{MaxSpec}(\mathcal{O}_K)}H^1\left(X_{\mathrm{\acute{e}t}},(i_\mathfrak{p})_\ast\underline{\mathbb{Z}}\right)\\ &=\bigoplus_{\mathfrak{p}\in\mathrm{MaxSpec}(\mathcal{O}_K)}H^1\left(\left(\mathrm{Spec}(\mathcal{O}_K/\mathfrak{p})\right)_{\mathrm{\acute{e}t}},\underline{\mathbb{Z}}\right)\end{aligned}

where the first isomorphism is because cohomology commutes with direct sums in this case, and the second since $i_\mathfrak{p}$ 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

\displaystyle \begin{aligned}H^1(\mathrm{Spec}(\mathcal{O}_k/\mathfrak{p}),\underline{\mathbb{Z}}) &= H^1(G_{\mathcal{O}_K/\mathfrak{p}},\mathbb{Z})\\ &= \mathrm{Hom}_{\mathrm{cont.}}(\widehat{\mathbb{Z}},\mathbb{Z})\\ &=0\end{aligned}

The computation of the second cohomology of $\mathrm{Div}_X$ succumbs to similar methods. Namely, the same reasoning shows that

$\displaystyle H^2(X_{\mathrm{\acute{e}t}},\mathrm{Div}_X)=\bigoplus_{\mathfrak{p}\in\mathrm{MaxSpec}(\mathcal{O}_K)}H^2(\widehat{\mathbb{Z}},\mathbb{Z})$

where the cohomology on the right hand side is continous cohomology with $\mathbb{Z}$ a trivial module.

The last relevant calculation we need to do is the computation of $H^2(X_{\mathrm{\acute{e}t}},\eta_\ast\mathbf{G}_{m,\mathrm{Spec}(K)})$. To do this we appeal to the Leray spectral sequence which gives us

$H^p\left(X_{\mathrm{\acute{e}t}},R^q\eta_\ast\mathbf{G}_{m,\mathrm{Spec}(K)}\right)\Rightarrow H^{p+q}((\mathrm{Spec}(K))_{\mathrm{\acute{e}t}},\mathbf{G}_{m,\mathrm{Spec}(K)})$

But, we claim that $R^q\eta_\ast\mathbf{G}_{m,\mathrm{Spec}(K)}=0$ for $q>0$. To do this, it suffices to check that it vanishes on geometric points centered at closed points. So, let $\overline{x}$ be a geometric point of $X$. Then, by standard theory we know that

$(R^q\eta_\ast\mathbf{G}_{m,\mathrm{Spec}(K)})_{\overline{x}}=H^q(\left(\mathrm{Spec}(K_{\overline{x}})_{\mathrm{\acute{e}t}}\right),\mathbf{G}_m)$

where $K_{\overline{x}}$ is the fraction field of strict henselization of $X$ at $\overline{x}$. One can verify that if $\overline{x}$ hits the point $\mathfrak{p}$ of $X$ then

$\displaystyle K_{\overline{x}}=W\left(\overline{\mathcal{O}_K/\mathfrak{p}}\right)\left[\frac{1}{p}\right]$

So, we need to prove that the Galois cohomology of $\mathbf{G}_m$ with respect to $K_{\overline{x}}$ is zero in positive degree. This is non-trivial. For $q=1$ this is Hilbert’s theorem 90. For $q=2$ 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 $K_{\overline{x}}$ is $1$, 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

$H^2(X_{\mathrm{\acute{e}t}},\eta_\ast\mathbf{G}_{m,\mathrm{Spec}(K)})=H^2(\left(\mathrm{Spec}(K)\right)_{\mathrm{\acute{e}t}},\mathbf{G}_m)=\mathrm{Br}(K)$

Thus, we have proven that there is an exact sequence

$\displaystyle 0\to \mathrm{Br}(\mathcal{O}_K)\to\mathrm{Br}(K)\to \bigoplus_{\mathfrak{p}}H^2(\widehat{\mathbb{Z}},\mathbb{Z})$

This seems relatively unhelpful since we don’t have a handle on this second map. The key observation though is that

$H^2(\widehat{\mathbb{Z}},\mathbb{Z})=\mathrm{Br}(K_\mathfrak{p})$

and that the diagram

$\begin{matrix}\mathrm{Br}(K) & \to & \displaystyle \bigoplus_{\mathfrak{p}}H^2(\widehat{\mathbb{Z}},\mathbb{Z})\\ {^\text{id}}\downarrow & & \downarrow^{\approx}\\ \mathrm{Br}(K) & \to & \displaystyle \bigoplus_{\mathfrak{p}}\mathrm{Br}(K_\mathfrak{p})\end{matrix}$

commutes. If this seems unbelievable, it’s because you’ve forgotten that this is precisely how one proves that $\mathrm{Br}(K_\mathfrak{p})=\mathbb{Q}/\mathbb{Z}$ (see, for example, Serre’s relevant article in Cassels and Frolich)!

Thus, we see that we can rewrite our sequence as

$\displaystyle 0\to\mathrm{Br}(\mathcal{O}_K)\to\mathrm{Br}(K)\to\bigoplus_{\mathfrak{p}}\mathrm{Br}(K_\mathfrak{p})$

But, combining this with the fundamental sequence of class field theory we obtain the following short exact sequence

$0\to \mathrm{Br}(\mathcal{O}_K)\to \left(\mathbb{Z}/2\mathbb{Z}\right)^{r}\to \left(\mathbb{Z}/2\mathbb{Z}\right)^\delta\to 0$

where $\delta$ is zero or $1$ depending on if $K$ has a real place or not. But, this sequence splits (it’s a sequence of $\mathbb{F}_2$-spaces!) and so we can finally conclude that

$\mathrm{Br}\left(\mathcal{O}_K\right)=\begin{cases}(\mathbb{Z}/2\mathbb{Z})^{r-1} & \mbox{if}\quad K\text{ has a real place}\\ 0 & \mbox{if}\quad \text{otherwise}\end{cases}$