Brauer group of $\mathbb{Z}_{(p)}$ This may be a well known result but I could not find it in the standard references. What is the Brauer group of the local ring $\mathbb{Z}_{(p)}$ (the ring of integers localized at $p$)?
 A: Lemma. Let $K$ be a number field with ring of integers $\mathcal O_K$, and let $S \subseteq \Omega_K^f$ be a set of finite places of $K$. Then there is a canonical short exact sequence
$$0 \to \operatorname{Br}(\mathcal O_{K,S}) \to \bigoplus_{v\ \in\ S\ \cup\ \Omega_K^\infty} \operatorname{Br}(K_v) \stackrel{\operatorname{inv}}\longrightarrow \mathbf Q/\mathbf Z,$$
whose final map is surjective if $S \neq \varnothing$.
Here $\mathcal O_{K,S}$ denotes the localisation of $\mathcal O_K$ away from $S$ (this notation is common when $S$ is a finite subset), meaning that the closed points of $\operatorname{Spec} \mathcal O_{K,S}$ are $\Omega_K^f \setminus S$.
Your result is the case $S = \Omega_K^f \setminus \{p\}$. The other extreme is $S = \varnothing$, which for number fields with at most $1$ real place (e.g. $K = \mathbf Q$, or any totally imaginary field) shows that $\operatorname{Br}(\mathcal O_K) = 0$.
Proof. For $S = \Omega_K^f$, this is the Albert–Brauer–Hasse–Noether theorem. Moreover, there is a short exact sequence
$$0 \to \mathbf G_m \to j_* \mathbf G_m \to \bigoplus_{v \in \Omega_K^f \setminus S} i_{v,*} \mathbf Z \to 0\tag{1}\label{1}$$
on the étale site of $\mathcal O_{K,S}$, where $j \colon \operatorname{Spec} K \hookrightarrow \operatorname{Spec} \mathcal O_{K,S}$ is the inclusion of the generic point, and $i_v \colon \operatorname{Spec} \kappa(v) \hookrightarrow \operatorname{Spec} \mathcal O_{K,S}$ the inclusion of the closed point $v$. Since $i_v$ is finite, the pushforward $i_{v,*}$ is exact, so
$$H^i\big(\operatorname{Spec} \mathcal O_{K,S}, i_{v,*}\mathbf Z\big) = H^i\big(\kappa(v), \mathbf Z\big) = \begin{cases} \mathbf Z & i = 0, \\ 0 & i = 1, \\ \mathbf Q/\mathbf Z & i = 2, \\ 0 & i > 2. \end{cases}$$
Write $K_{(\bar v)}^{\operatorname{sh}}$ for the fraction field of the strict Henselisation $\mathcal O_{K,\bar v}^{\operatorname{sh}}$ of $\mathcal O_K$ at a geometric point $\bar v \to \operatorname{Spec} \mathcal O_{K,S}$ (the parentheses $(\bar v)$ distinguish it more clearly from the completion $K_v$). Then the stalk of $R^ij_*\mathbf G_m$ at $\bar v$ is $H^i(K_{(v)}^{\operatorname{sh}},\mathbf G_m)$, which vanishes for $i > 0$ by a version of Lang's theorem (according to Milne [Milne, Example III.2.22(b)], this version is somewhere in [Shatz]). We conclude that $R^ij_*\mathbf G_m = 0$ for $i > 0$ and therefore $$H^i(\operatorname{Spec} \mathcal O_{K,S}, j_*\mathbf G_m) = H^i(K,\mathbf G_m)$$ for all $i$. Since cohomology commutes with direct sums, \eqref{1} gives an exact sequence
$$0 \to H^2(\operatorname{Spec} \mathcal O_{K,S},\mathbf G_m) \to H^2(\operatorname{Spec} K,\mathbf G_m) \to \bigoplus_{v \in \Omega_K^f \setminus S} \mathbf Q/\mathbf Z,\tag{2}\label{2}$$
where the last map is the usual invariant map. Thus the result for $\mathcal O_{K,S}$ follows from the result for $K$, since we get exactly the elements of $\bigoplus_{v\in \Omega_K} \operatorname{Br}(K_v)$ whose coordinates at all $v \in \Omega_K^f \setminus S$ vanish. $\square$
The slogan of \eqref{2} is that $\operatorname{Br}(\mathcal O_{K,S})$ consists of those elements of $\operatorname{Br}(K)$ that are unfamified at all closed points of $\operatorname{Spec} \mathcal O_{K,S}$. This should be a general principle when removing divisors.

References.
[Milne]  J. S. Milne, Étale cohomology. Princeton Mathematical Series 33. Princeton University Press, 1980. ZBL0433.14012.
[Shatz]  S. S. Shatz, Profinite groups, arithmetic, and geometry, Annals of Mathematics Studies 67. Princeton University Press, 1972. ZBL0236.12002, jstor.
A: As another perspective, there is an exact sequence
$$0 \to Br(\mathbb Z_{(p)}) \to Br(\mathbb Q) \to \mathbb Q/Z \to 0$$
where the map on the right is given by the ramification (aka Hasse invariant) at $p$ (similarly for more general global fields).
I would like to say that this is in Saltman's Lectures on Division Algebras, as it is essentially Theorem 10.3 there. Unfortunately, this is only done there in the case that the order of the Brauer class is prime to the residue characteristic. Saltman does it like this because for more general fields, ramification can be a bit more complicated in the $p,p$ case.
On the other hand, the only place this is used is in the existence of an unramified splitting field for Brauer classes at the completion. Over local fields this can be found, for example, in Serre's Local Fields, Chapter 12. So, in fact, the argument in Saltman's book works in general for the glocal field case.
