Lemma 42.28.3 (02TJ)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.28: Intersecting with an invertible sheaf and rational equivalence
Lemma 42.28.3 (cite)

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Lemma 42.28.3. 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 invertible on $X$ . For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _{k + 2}(X)$ we have

$c_1(\mathcal{L}) \cap c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{N}) \cap c_1(\mathcal{L}) \cap \alpha$

as elements of $\mathop{\mathrm{CH}}\nolimits _ k(X)$ .

Proof. Write $\alpha = \sum m_ j[Z_ j]$ for some locally finite collection of integral closed subschemes $Z_ j \subset X$ with $\dim _\delta (Z_ j) = k + 2$ . Consider the proper morphism $p : \coprod Z_ j \to X$ . Set $\alpha ' = \sum m_ j[Z_ j]$ as a $(k + 2)$ -cycle on $\coprod Z_ j$ . By several applications of Lemma 42.26.4 we see that $c_1(\mathcal{L}) \cap c_1(\mathcal{N}) \cap \alpha = p_*(c_1(p^*\mathcal{L}) \cap c_1(p^*\mathcal{N}) \cap \alpha ')$ and $c_1(\mathcal{N}) \cap c_1(\mathcal{L}) \cap \alpha = p_*(c_1(p^*\mathcal{N}) \cap c_1(p^*\mathcal{L}) \cap \alpha ')$ . Hence it suffices to prove the formula in case $X$ is integral and $\alpha = [X]$ . In this case the result follows from Lemma 42.28.1 and the definitions. $\square$

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