Lemma 42.28.1 (02TH)—The Stacks project

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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 42.28.1 (cite)

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Lemma 42.28.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim _\delta (X) = n$. Let $\mathcal{L}$, $\mathcal{N}$ be invertible on $X$. Choose a nonzero meromorphic section $s$ of $\mathcal{L}$ and a nonzero meromorphic section $t$ of $\mathcal{N}$. Set $\alpha = \text{div}_\mathcal {L}(s)$ and $\beta = \text{div}_\mathcal {N}(t)$. Then

\[ c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L}) \cap \beta \]

in $\mathop{\mathrm{CH}}\nolimits _{n - 2}(X)$.

Proof. Immediate from the key Lemma 42.27.1 and the discussion preceding it. $\square$

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