Brauer groups and étale homotopy type

Abstract

Extending a result of Schröer on a Grothendieck question in the context of complex analytic spaces, we prove that the surjectivity of the Brauer map $\delta :Br(X)\to H_{\acute{\mathrm{e}}\mathrm{t}}^2(X,{\mathbb{G}}_{m,X}{)}_{\mathrm{tor}}$ for algebraic schemes depends on their étale 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.

27 pages, comments are welcome