Brauer groups and étale homotopy type

Mohammed Moutand

Mathematische Nachrichten, 2024, vol. 297, issue 1, 229-245

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 δ:Br(X)→Hét2(X,Gm,X)tors$\delta : \operatorname{Br}(X) \rightarrow H_{{\text{\rm \'{e}t}}}^2(X, \mathbb {G}_{m, X})_{\rm tors}$ for schemes depends on their étale homotopy type. We use properties of algebraic K(π,1)$K(\pi , 1)$ 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.

Date: 2024
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https://doi.org/10.1002/mana.202200190

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