A note on the Brauer group of Laurent polynomial rings

Abstract

In this article, we compute the Brauer group of Laurent polynomial rings. We show that if X is a smooth quasi-projective scheme over an algebraically closed field k of characteristic zero, then

$$\begin{aligned} {\text {Br}}\left( X\times _{\textrm{Spec} (k)} \textrm{Spec} \left( k\left[ t_{1}, t_{1}^{-1}, t_2, t_{2}^{-1}, \dots , t_{n}, t_{n}^{-1}\right] \right) \right) \end{aligned}$$

admits a natural decomposition, more explicitly, it is the direct sum of ${\text {Br}}({X})$, n copies of $H_{et}^{1}(X, {\mathbb {Q}}/{\mathbb {Z}})$, and $n(n-1)/2$ copies of $H_{et}^{0}(X, {\mathbb {Q}}/{\mathbb {Z}}).$ We also discuss the case when X may not be smooth.