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Let $X \to \mathbb{A}^1_k$ be a smooth morphisms of varieties over a field $k$. Its generic fiber is a smooth morphism $X_\eta \to \eta = \text{Spec }k(t)$. Is it true that we have an injection

$H^2(X_{et},\mathbb{G}_m) \to H^2((X_\eta)_{et},\mathbb{G}_m)$?

Where does it come from and why does this hold?

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1 Answer 1

up vote 2 down vote accepted

For a regular, integral, quasi-compact scheme $X$, we have $H^2(X_{et}, \mathbb{G}_m) \hookrightarrow H^2(K(X)_{et},\mathbb{G}_m)$ (use the Leray spectral sequence for the inclusion of the generic point), and you can factor this injection into $H^2(X_{et}, \mathbb{G}_m) \to H^2((X_\eta)_{et}, \mathbb{G}_m) \hookrightarrow H^2(K(X)_{et},\mathbb{G}_m)$.

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