Let $R$ be a torsion-free regular noetherian ring. The Brauer group $Br(R)$ of $R$, defined equivalently (by a [theorem of Gabber](http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf%E2%80%8E)) as the group of Morita equivalence classes of Azumaya $R$-algebras or as the étale cohomology $H^2(X\_{et}, \mathbb{G}\_m)$, has the property that it is homotopy invariant: that is, $Br(R) \simeq Br( R[t])$. (I learned this from Auslander and Goldman's [article](http://www.ams.org/tran/1960-097-03/.../S0002-9947-1960-0121392-6.pdf%E2%80%8E) "The Brauer group of a commutative ring." )


Auslander and Goldman's argument is that $Br(R)$ is always a summand of $Br(R[t])$ (because of the natural retraction), so it suffices to show this homotopy invariance property when $R$ is a field $k$ of characteristic zero: this is because Brauer groups of regular domains inject into those of their quotient fields. In this case, 
one can use Galois descent along $k \to \bar{k}$ together with "Tsen's theorem." (The reason for characteristic zero is simply that Tsen's theorem requires an algebraically closed, rather than separably closed, base field.)


Now this isn't true when the affine line is replaced by a punctured affine line, for instance, $\mathbb{G}\_m$ or $\mathbb{A}^1 \setminus \{0, 1\}$: even the Galois descent argument (when $R$ is a field) already breaks down, because the units in the ring of functions on $R[t, t^{-1}]$ is not simply $R^{\times}$, but 
rather $R^{\times} \oplus \mathbb{Z}$. In particular, if $R = k$ is a field, this will lead to a $H^2( G\_k, \mathbb{Z})$ (which is the Pontryagin dual of the Galois group $G\_k$) sitting inside the Brauer group of $k[t, t^{-1}]$. 


I'm curious if there is a general procedure for calculating the Brauer group of a punctured affine line over a regular ring in terms of the Brauer group of the 
base ring. (I have in mind something like a localization of $\mathbb{Z}$ as the base.)



Brauer groups of punctured affine lines over a base
This is a calculation using the local cohomology sequence for etale cohomology together with
Gabber's relatively recent proof of the absolute purity conjecture. Let $%
Y=\operatorname{Spec}(R)$, $X=\operatorname{Spec}( R[t]) $, and let $\sigma \subseteq X$ be the
image of a section so that $U=\operatorname{Spec}\left( R[t,t^{-1}]\right) $ is its
complement. Gabber's recent proof of absolute purity establishes
Grothendieck's purity conjecture for the Brauer group which states that $%
H\_{\sigma }^{3}\left( X,\mathbb{G}\_{m}\right) =H^{1}\left( \sigma ,\mathbb{Q}/\mathbb{%
Z}\right) $ if $\sigma \subseteq X$ is a regular subscheme of a regular
scheme. Moreover $R\rightarrow R[t,t^{-1}]$ has a section. Thus we have an
exact sequence coming from the local cohomology of $\mathbb{G}\_{m},$
\begin{equation\*}
0\rightarrow Br\left( R\right) \rightarrow Br\left( R[t,t^{-1}]\right)
\rightarrow H^{1}\left( R,\mathbb{Q}/\mathbb{Z}\right) \rightarrow 0
\end{equation\*}
since $H^{3}\left( R,\mathbb{G}\_{m}\right) =H^{3}\left( R[t],\mathbb{G}\_{m}\right)
\hookrightarrow H^{3}(R[t,t^{-1}],\mathbb{G}\_{m}).$ Note though that this uses Kummer
sequences and homotopy invariance for $\mu\_n$ and so only applies to torsion in $Br(R[t,t^{-1}])$ of order
invertible in $R.$ In fact you can argue directly with $n$ torsion using Kummer sequences and purity for $\mu\_n$ cohomology to get this result for $n$ torsion elements if $n$ is a unit in $R$.


Of course if you puncture the line in more than one point, you get multiple copies of $ H^{1}\left( R,\mathbb{Q}/\mathbb{Z}\right)$.


The easiest way to address $n$ torsion when $n$ is not a unit in $R$ is, I think, to look at the split exact sequence on $R\_{et}$
$$
0\rightarrow k\_\*\mathbb{G}\_{m,R[t]} \rightarrow p\_\*\mathbb{G}\_{m,R[t,t^{-1}]} \rightarrow \mathbb{Z} \rightarrow 0
$$
where the last map is the degree map. The Leray spectral sequence for $k$ and $p$ yield a split exact sequence
$$
0\rightarrow Br(R) \rightarrow Br\_{sp}(R[t,t^{-1}]) \rightarrow H^1(R,\mathbb{Q}/\mathbb{Z})\rightarrow 0
$$
where $Br\_{sp}(R[t,t^{-1}])$ consists of elenents in $Br(R[t,t^{-1}])$ that are split by passing to $R\_{\mathfrak{p}}^{sh}[t,t^{-1}]$ as $R\_{\mathfrak{p}}^{sh}$ runs through all strict henselizations of $R$ at prime ideals $\mathfrak{p}$.


