The Stacks project

Lemma 115.23.5. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a proper morphism. Let $D \subset Y$ be an effective Cartier divisor. Assume $X$, $Y$ integral, $n = \dim _\delta (X) = \dim _\delta (Y)$ and $f$ dominant. Then

\[ f_*[f^{-1}(D)]_{n - 1} = [R(X) : R(Y)] [D]_{n - 1}. \]

In particular if $f$ is birational then $f_*[f^{-1}(D)]_{n - 1} = [D]_{n - 1}$.

Proof. Immediate from Chow Homology, Lemma 42.26.3 and the fact that $D$ is the zero scheme of the canonical section $1_ D$ of $\mathcal{O}_ X(D)$. $\square$

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