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I would like to show that for smooth, nice variety $X$, $Pic(X \times A^{1})=Pic(X)$.

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.

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This is Proposition II.6.6 in Hartshorne. I'm not sure if the proof can be written down without schemes. – Martin Brandenburg Mar 26 '11 at 15:39
What you've written under "surjectivity" seems to actually be "injectivity". – Matt Mar 26 '11 at 20:44

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