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For questions requesting examples of a certain structure or phenomenon

8 votes

Gerbes for a cyclic group. (or maybe G_m too)

In this case, $H^1(X, \mathbb{G}_m)=0$ by Hilbert's Theorem 90, and yet there are plenty of examples where $Br(X)$ is very much non-trivial. …
Anatoly Preygel's user avatar