The Stacks project

Lemma 81.18.4. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $Y \subset X$ be a closed subscheme with $\dim _\delta (Y) \leq k + 1$ and let $s \in \Gamma (Y, \mathcal{L}|_ Y)$ be a regular section. Then

\[ c_1(\mathcal{L}) \cap [Y]_{k + 1} = [Z(s)]_ k \]

in $\mathop{\mathrm{CH}}\nolimits _ k(X)$.

Proof. Write

\[ [Y]_{k + 1} = \sum n_ i[Y_ i] \]

where $Y_ i \subset Y$ are the irreducible components of $Y$ of $\delta $-dimension $k + 1$ and $n_ i > 0$. By assumption the restriction $s_ i = s|_{Y_ i} \in \Gamma (Y_ i, \mathcal{L}|_{Y_ i})$ is not zero, and hence is a regular section. By Lemma 81.17.2 we see that $[Z(s_ i)]_ k$ represents $c_1(\mathcal{L}|_{Y_ i})$. Hence by definition

\[ c_1(\mathcal{L}) \cap [Y]_{k + 1} = \sum n_ i[Z(s_ i)]_ k \]

Thus the result follows from Lemma 81.18.3. $\square$

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