Lemma 82.22.5 (0ER5)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 82: Chow Groups of Spaces
Section 82.22: Intersecting with effective Cartier divisors
Lemma 82.22.5 (cite)

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Lemma 82.22.5. In Situation 82.2.1. Let $f : X' \to X$ be a proper morphism of good algebraic spaces over $B$ . Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 82.22.1. Form the diagram

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

as in Remark 82.22.3. For any $(k + 1)$ -cycle $\alpha '$ on $X'$ we have $i^*f_*\alpha ' = g_*(i')^*\alpha '$ in $\mathop{\mathrm{CH}}\nolimits _ k(D)$ (this makes sense as $f_*$ is defined on the level of cycles).

Proof. Suppose $\alpha = [W']$ for some integral closed subspace $W' \subset X'$ . Let $W \subset X$ be the “image” of $W'$ as in Lemma 82.7.1. In case $W' \not\subset D'$ , then $W \not\subset D$ and we see that

$[W' \cap D']_ k = \text{div}_{\mathcal{L}'|_{W'}}({s'|_{W'}}) \quad \text{and}\quad [W \cap D]_ k = \text{div}_{\mathcal{L}|_ W}(s|_ W)$

and hence $f_*$ of the first cycle equals the second cycle by Lemma 82.19.3. Hence the equality holds as cycles. In case $W' \subset D'$ , then $W \subset D$ and $f_*(c_1(\mathcal{L}|_{W'}) \cap [W'])$ is equal to $c_1(\mathcal{L}|_ W) \cap [W]$ in $\mathop{\mathrm{CH}}\nolimits _ k(W)$ by the second assertion of Lemma 82.19.3. By Remark 82.15.3 the result follows for general $\alpha '$ . $\square$

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