Lemma 42.34.3 (0B77)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.34: Chow cohomology and the first Chern class
Lemma 42.34.3 (cite)

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Lemma 42.34.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then the rule that to $f : X' \to X$ assigns $c_1(f^*\mathcal{L}) \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \to \mathop{\mathrm{CH}}\nolimits _{k - 1}(X')$ is a bivariant class of degree $1$.

Proof. This follows from Lemmas 42.28.2, 42.26.4, 42.26.2, and 42.30.4. $\square$

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