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For a variety $X$ over a field $k$, we define $Br(X) = H^2_{et}(X,\mathbb{G}_m)$.

Suppose $X$ is a smooth variety (finite type, separated) over an algebraically closed field $k$ together with a morphism $X \to \mathbb{A}^1_k$. Let $X_{\eta}$ be the generic fiber of this morphism, which is a smooth variety over $k(t)$.

Is it true that there is always an injection $Br(X) \to Br(X_\eta)$? How is it constructed, why do you need smoothness to get injectivity (if this is true, of course)?

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Any update on this? I'm curious to know the answer. I don't know a lot about Brauer groups, but it seems possible in my mind that you could have a non-trivial Brauer class that is generically trivial (and hence not have an injection), but smoothness might rule this out. –  Matt Dec 2 '12 at 6:34

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