The Stacks project

Lemma 42.18.1. 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 $n = \dim _\delta (X) = \dim _\delta (Y)$. Let $p : X \to Y$ be a dominant proper morphism. Let $f \in R(X)^*$. Set

\[ g = \text{Nm}_{R(X)/R(Y)}(f). \]

Then we have $p_*\text{div}(f) = \text{div}(g)$.

Proof. Let $Z \subset Y$ be an integral closed subscheme of $\delta $-dimension $n - 1$. We want to show that the coefficient of $[Z]$ in $p_*\text{div}(f)$ and $\text{div}(g)$ are equal. We may apply Lemma 42.16.2 to the morphism $p : X \to Y$ and the generic point $\xi \in Z$. Hence we may replace $Y$ by an affine open neighbourhood of $\xi $ and assume that $p : X \to Y$ is finite. Write $Y = \mathop{\mathrm{Spec}}(R)$ and $X = \mathop{\mathrm{Spec}}(A)$ with $p$ induced by a finite homomorphism $R \to A$ of Noetherian domains which induces an finite field extension $L/K$ of fraction fields. Now we have $f \in L$, $g = \text{Nm}(f) \in K$, and a prime $\mathfrak p \subset R$ with $\dim (R_{\mathfrak p}) = 1$. The coefficient of $[Z]$ in $\text{div}_ Y(g)$ is $\text{ord}_{R_\mathfrak p}(g)$. The coefficient of $[Z]$ in $p_*\text{div}_ X(f)$ is

\[ \sum \nolimits _{\mathfrak q\text{ lying over }\mathfrak p} [\kappa (\mathfrak q) : \kappa (\mathfrak p)] \text{ord}_{A_{\mathfrak q}}(f) \]

The desired equality therefore follows from Algebra, Lemma 10.121.8. $\square$

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