Lemma 42.25.4. 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. Let $Y \subset X$ be a closed subscheme. Let $s \in \Gamma (Y, \mathcal{L}|_ Y)$. Assume

$\dim _\delta (Y) \leq k + 1$,

$\dim _\delta (Z(s)) \leq k$, and

for every generic point $\xi $ of an irreducible component of $Z(s)$ of $\delta $-dimension $k$ the multiplication by $s$ induces an injection $\mathcal{O}_{Y, \xi } \to (\mathcal{L}|_ Y)_\xi $

^{1}.

Then

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

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