Lemma 42.32.7. In Situation 42.7.1 let $X$ be a scheme locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition 42.29.1. There exists a commutative diagram

such that

$p$ and $g$ are of finite type and flat of relative dimension $1$,

$p^* : \mathop{\mathrm{CH}}\nolimits _ k(D) \to \mathop{\mathrm{CH}}\nolimits _{k + 1}(D')$ is injective for all $k$,

$D' \subset X'$ is the zero scheme of a global section $s' \in \Gamma (X', \mathcal{O}_{X'})$,

$p^*i^* = (i')^*g^*$ as maps $\mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(D')$.

Moreover, these properties remain true after arbitrary base change by morphisms $Y \to X$ which are locally of finite type.

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