Lemma 42.17.2 (02RR)—The Stacks project

Lemma 42.17.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ , $Y$ be locally of finite type over $S$ . Assume $X$ , $Y$ are integral and $n = \dim _\delta (Y)$ . Let $f : X \to Y$ be a flat morphism of relative dimension $r$ . Let $g \in R(Y)^*$ . Then

$f^*(\text{div}_ Y(g)) = \text{div}_ X(g)$

in $Z_{n + r - 1}(X)$ .

Proof. Note that since $f$ is flat it is dominant so that $f$ induces an embedding $R(Y) \subset R(X)$ , and hence we may think of $g$ as an element of $R(X)^*$ . Let $Z \subset X$ be an integral closed subscheme of $\delta$ -dimension $n + r - 1$ . Let $\xi \in Z$ be its generic point. If $\dim _\delta (f(Z)) > n - 1$ , then we see that the coefficient of $[Z]$ in the left and right hand side of the equation is zero. Hence we may assume that $Z' = \overline{f(Z)}$ is an integral closed subscheme of $Y$ of $\delta$ -dimension $n - 1$ . Let $\xi ' = f(\xi )$ . It is the generic point of $Z'$ . Set $A = \mathcal{O}_{Y, \xi '}$ , $B = \mathcal{O}_{X, \xi }$ . The ring map $A \to B$ is a flat local homomorphism of Noetherian local domains of dimension $1$ . We have $g$ in the fraction field of $A$ . What we have to show is that

$\text{ord}_ A(g) \text{length}_ B(B/\mathfrak m_ AB) = \text{ord}_ B(g).$

This follows from Algebra, Lemma 10.52.13 (details omitted). $\square$

The Stacks project

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