Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$

Question

Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from class field theory.

The original poster of this question is more specific, stating that it follows from the Albert-Brauer-Hasse-Noether (ABHN) theorem. My understanding of the ABHN theorem is that we have the short exact sequence $$0\to\mathrm{Br}(K)\to\bigoplus_v \mathrm{Br}(K_v)\to\mathbb{Q}/\mathbb{Z}\to 0,$$ for any global field $K$ (with $v$ running over every place of $K$).

In Grothendieck's LE GROUPE DE BRAUER III: EXEMPLES ET COMPLEMENTS, he uses the ABHN theorem to treat the case $X=\mathrm{Spec}(R)$ when $R$ is the ring of integers in a number field $K$. But then, in Remark (2.5) c), he says that the finiteness of $\mathrm{Br}(X)$ (in our more general case) follows from the preceding results... How exactly?

Answer 1

Let me try for $X = Spec(\mathbb{Z})$ ( $K = \mathbb{Q}$ ) to illustrate the idea. Since by ABHN, we have $0 \rightarrow Br(K) \xrightarrow{i} \bigoplus_v Br(K_v) \xrightarrow{p} \mathbb{Q}/\mathbb{Z}\to 0$. Since $Br(X) \hookrightarrow Br(K)$. Hence using this inclusion, it is enough to show the image of $Br(X) $ in $\bigoplus_v Br(K_v)$ is finite (in this case it would be zero). First note that its image in $Br(K_{v})$ is zero for each finite place $v$ as this factor through $Br(\mathbb{Z}_{v})$ which is zero as the residue field of $\mathbb{Z}_{v}$ is finite. So you only have to worry about the image of $Br(X)$ inside $Br(K)_{v}$ for the infinite place $v$, but the group $Br(K)_{v} = Br(\mathbb{R}) = \mathbb{Z}/2$, which is finite, which proves the claim. Infact this image in this case is zero as the map $p$ is injective when restricted to $Br(\mathbb{R})$.

Now the above argument can be mimicked to any regular curve which is proper over $Spec(\mathbb{Z})$ as each argument will go through the same as above except you need to be careful about the images in non-finite places. Here you may not be able to deduce vanishing but certainly finiteness which you wanted.