Lemma 42.33.4 (0EPK)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.33: Bivariant intersection theory
Lemma 42.33.4 (cite)

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Lemma 42.33.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes locally of finite type over $S$ . Let $c \in A^ p(X \to Z)$ and assume $f$ is proper. Then the rule that to $Z' \to Z$ assigns $\alpha \longmapsto f'_*(c \cap \alpha )$ is a bivariant class denoted $f_* \circ c \in A^ p(Y \to Z)$ .

Proof. This follows from Lemmas 42.12.2, 42.15.1, and 42.29.8. $\square$

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