Lemma 42.32.7 (0F99)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.32: Affine bundles
Lemma 42.32.7 (cite)

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

\[ \xymatrix{ D' \ar[r]_{i'} \ar[d]_ p & X' \ar[d]^ g \\ D \ar[r]^ i & X } \]

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.

Proof. Observe that $(i')^*$ is defined because we have the triple $(\mathcal{O}_{X'}, s', i' : D' \to X')$ as in Definition 42.29.1. Thus the statement makes sense.

Set $\mathcal{L}_1 = \mathcal{O}_ X$, $\mathcal{L}_2 = \mathcal{L}$ and apply Lemma 42.32.5 with the section $s$ of $\mathcal{L} = \mathcal{L}_1 \otimes _{\mathcal{O}_ X} \mathcal{L}_2$. Take $D' = D_1$. The results now follow from the lemma, from Lemma 42.32.6 and injectivity by Lemma 42.32.2. $\square$

The Stacks project

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