TITLE: Picard groups of (fiber) products
QUESTION [6 upvotes]: Let us work in the "nice" situation where $X,Y,Z$ are smooth complex algebraic varieties, not necessarily compact. Assume that the fiber product $W:= X \times_Z Y$ is also smooth. What can we say about the Picard group of $W$? 
More precisely, assume that $Pic(X)=Pic(Z)=0$.
1) Can we deduce that $Pic(W)=Pic(Y)$?
2) If 1) is not true in general, can we draw the conclusion if $Z$ is a point and thus $W=X \times Y$?
3) If 1) is not true in general, can we draw the conclusion if $Y \to Z$ is a finite étale cover?

REPLY [5 votes]: A weakening of 2) is an exercise (III.12.6) in Hartshorne: Let $X$ be an integral projective scheme over an algebraically closed field $k$ and assume that $H^1(X, \mathcal O_X) = 0$. Let $T$ be a connected scheme of finite type over $k$. Then $\textrm{Pic}(X) \times \textrm{Pic}(T) \cong \textrm{Pic}(X \times T) $ under the obvious morphism. However, it is not true in general that $\textrm{Pic}(X) \times \textrm{Pic}(T) \cong \textrm{Pic}(X \times T) $ for two arbitrary $k$-schemes $X$ and $T$, cf exercise IV.4.10 in Hartshorne.
In general, you might be interested in exercises III.12.4 and III.12.5 of Hartshorne as well; they give more results on Picard groups in this context.