I would like to show that for smooth, nice variety $X$, $\operatorname{Pic}(X \times A^{1})=\operatorname{Pic}(X)$.
I only need to prove this in the generality where the Picard group and the divisor class group are the same.
The pullback homomorphism $p^{*}:Cl(X)\rightarrow Cl(X \times A^{1})$ of the projection is my best guess for an isomorphism.
Injectivity: Let $D$ be a divisor of $X$. $p^{*}(D)=(f) \in K(t)$. We need to show that $f$ is in $K$, but I am not sure how to do this.
Surjectivity: Here I have less of an idea. Is it enough to show that if $D_{1}$ and $D_{2}$ get sent to the same thing then there is an $f$ such that $D_{1}-D_{2}=div(f)$?
Disclaimer: I do not know about schemes.