Let $X$ be a noetherian normal scheme, let $\mathbb{A}^1$ be the affine line over $\mathbb{Z}$, and let $X \times \mathbb{A}^1$ be the fibre product over $\mathbb{Z}$ with projection $$ \pi: X \times \mathbb{A^1} \longrightarrow X . $$
In Chapter II of Hartshorne, he shows that the Weil divisor class groups satisfy $$ \text{Cl}(X \times \mathbb{A}^1) \simeq \text{Cl}(X).$$ To do this he defines a group homomorphism, $$\Psi: \text{Cl}(X) \rightarrow \text{Cl}(X \times \mathbb{A}^1) $$ by $\Psi(Z) = \pi^{-1}(Z)$.
My question is why $\pi^{-1}$ should even be a Weil divisor. In particular, why does it have codimension $1$ in $X \times \mathbb{A}^1$?
I think this can be done by appealing to the faithful flatness of $\pi$, but at that point in Hartshorne no such notion has been introduced or even mentioned. So I am wondering if there is some easy way to see it in this case?