Let $X$ be a noetherian integral separated scheme which is regular in codimension one. Then the proposition states that $\operatorname{Cl}(X)\cong\operatorname{Cl}(X\times\mathbb{A}^1_\mathbb{Z})$, where $\operatorname{Cl}$ denotes the divisor class group. In the proof, the author defines a homomorphism $\operatorname{Cl}(X)\to\operatorname{Cl}(X\times\mathbb{A}^1_\mathbb{Z})$ by sending a prime divisor $Y$ on $X$ to $\pi^{-1}(Y)$ on $X\times\mathbb{A}^1_\mathbb{Z}$.
Why is $\pi^{-1}(Y)$ a prime divisor?
Reducing to affine open sets, it suffices to prove that if $A$ is a noetherian integral domain (regular in codimension one) and $\mathfrak{p}$ is a prime ideal of $A$ with height $1$, then $\mathfrak{p}[t]$ is a prime ideal of $A[t]$ with height $1$. How do I prove this? Why can't there be a prime $\mathfrak{q}$ of $A[t]$ with $0\subsetneq\mathfrak{q}\subsetneq\mathfrak{p}[t]$ and $\mathfrak{q}\cap A=0$?
