Lemma 42.25.2 (02SP)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.25: Intersecting with an invertible sheaf
Lemma 42.25.2 (cite)

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Lemma 42.25.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$, $\mathcal{N}$ be an invertible sheaves on $X$. Then

\[ c_1(\mathcal{L}) \cap \alpha + c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}) \cap \alpha \]

in $\mathop{\mathrm{CH}}\nolimits _ k(X)$ for every $\alpha \in Z_{k + 1}(X)$. Moreover, $c_1(\mathcal{O}_ X) \cap \alpha = 0$ for all $\alpha $.

Proof. The additivity follows directly from Divisors, Lemma 31.27.5 and the definitions. To see that $c_1(\mathcal{O}_ X) \cap \alpha = 0$ consider the section $1 \in \Gamma (X, \mathcal{O}_ X)$. This restricts to an everywhere nonzero section on any integral closed subscheme $W \subset X$. Hence $c_1(\mathcal{O}_ X) \cap [W] = 0$ as desired. $\square$

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