  Extending a result of Schr\"oer on a Grothendieck question in the context of
complex analytic spaces, we prove that the surjectivity of the Brauer map
$\delta: Br(X) \rightarrow H_{\rm \'et}^2(X,\mathbb{G}_{m, X})_{\rm tor}$ for
algebraic schemes depends on their \'etale homotopy type. We use properties of
algebraic $K(\pi, 1)$ spaces to apply this to some classes of proper and smooth
algebraic schemes. In particular we recover a result of Hoobler and Berkovich
for abelian varieties. Further, we give an additional condition for the
surjectivity of $\delta$ which involves pro-universal covers. All proposed
conditions turn out to be equivalent for smooth quasi-projective varieties.
