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?



Brauer group and smoothness
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 $H^2(X\_{et}, \mathbb{G}\_m) \to H^2((X\_\eta)\_{et}, \mathbb{G}\_m) \hookrightarrow H^2(K(X)\_{et},\mathbb{G}\_m)$.


