Let $R$ be a torsion-free regular noetherian ring. The Brauer group $Br(R)$ of $R$, defined equivalently (by a theorem of Gabber) 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 "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.)

notalgebraically closed of char. $p > 0$ then ${\rm{Br}}(k(t))[p]$ is huge. It suffices (Ch.II, sec.3, Prop.5 in Serre's "Galois cohomology") to make degree-$p$ Galois extensions $L$ of $K=k(t)$ such that the norm $L^{\times}\rightarrow K^{\times}$ is not surjective. For $L$ wild over $\infty$, the norm between those residue fields is $p$-power on $k^{\times}$, so $f$ integral at $\infty$ with $f(\infty)\not\in k^p$ isn't a norm. $\endgroup$6more comments