I'm trying to understand the following proposition, which is Hartshorne Ⅱ prop 6.6.
Let $X$ be a noetherian integral separated scheme which is regular in codimension one, then $X \times_{\operatorname{Spec}\mathbb{Z}} \operatorname{Spec}\mathbb{Z}[t]$ is also a noetherian integral separated scheme which is regular in codimension one and $\operatorname{Cl} X\cong \operatorname{Cl}(X\times_{\operatorname{Spec}\mathbb{Z}}\operatorname{Spec}\mathbb{Z}[t])$.
I showed when $X$ is affine. I'm in trouble in general case.
Let $Y$ be a prime divisor of $X$, then $\pi^{-1}(Y)$ should be a prime divisor of $X\times_{\operatorname{Spec}\mathbb{Z}}\operatorname{Spec}\mathbb{Z}[t]$. But I can't show that $\pi^{-1}(Y)$ is irreducible. $\pi$ is a projetion $X \times_{\operatorname{Spec}\mathbb{Z}} \operatorname{Spec}\mathbb{Z}[t] \rightarrow X$. I know $X \times_{\operatorname{Spec}\mathbb{Z}} \operatorname{Spec}\mathbb{Z}[t]$ is irreducible, so this can be used.