I am reading the book ‘An Introduction to Birational Geometry of Algebraic Varieties’ by Shigeru Iitaka. In section 2.11 we attempt to define the pullback of a prime divisor. In particular, we have the following:
Suppose $f:V\rightarrow Z$ is a morphism of varieties where $V$ is a normal variety and $Z$ is locally factorial. (Here, locally factorial means all local rings are UFDs).
Suppose $\Gamma$ is a prime divisor on $Z$ such that $f(V) \not\subset \Gamma$ (so that $f^{-1}(\Gamma)$ is the not the whole of $V$) and also $f(V) \cap \Gamma \neq \emptyset$ (so that $f^{-1}(\Gamma)$ is not empty.)
He then proceed to claim the following: “By theorem 2.7, $f^{-1}(\Gamma)$ is union of (finitely many) prime divisors $W_1,\cdots, W_r$ of $V$. For context, here is theorem 2.7 of the book:
Theorem 2.7: Let $R$ be a Noetherian integral domain and $f$ be a nonzero and nonunit element in $R$. Then every minimal prime ideal of $fR$ is of height 1; hence, $\operatorname{ht}(fR)=1$.
My question is: How do we use theorem 2.7 to conclude that the preimage $f^{-1}(\Gamma)$ of a prime divisor must be union of finitely many prime divisors?
Any help given would be greatly appreciated! Thanks!
I gave this question some thought and I think I may have found out why. (Although there are still some parts which I am not entirely sure) Anyway here goes:
Given a morphism $f:V\rightarrow Z$ satisfying hypothesis as above, since $Z$ is locally factorial, any prime divisor $\Gamma$ on $Z$ is locally principal, i.e. we have an affine open cover $\{U_\lambda:= \operatorname{Spec}A_\lambda\}_\lambda$ such that $V_\lambda := \Gamma\cap U_\lambda$ can be written as $V(\varphi_\lambda)$ for some element $\varphi_\lambda \in A_\lambda$.
We choose an affine open subset $\operatorname{Spec} B_\lambda$ of $V$ mapping into $U_\lambda$ by $f$, i.e. we consider the restriction $f: \operatorname{Spec}B_\lambda \rightarrow \operatorname{Spec}A_\lambda$. This in turn gives a ring homomorphism $\theta: A_\lambda \rightarrow B_\lambda$.
We know that by the properties of $f$ and $\theta$, one has $f^{-1}(V(\varphi_\lambda)) = V(\theta(\varphi_\lambda) B_\lambda)$.
Now, by the theorem 2.7, we have every minimal prime ideal of $\theta(\varphi_\lambda) B_\lambda$ is of height 1. Hence coupled with the fact that $B_\lambda$ is Noetherian, we have $f^{-1}(V(\varphi_\lambda))$ is a finite union of $V(\mathfrak{q}_\lambda)$’s where each $\mathfrak{q}_\lambda$ are height 1 prime ideals in $B_\lambda$.
Hence locally, the statement holds. How can I translate statement to the global case?