Lemma 42.29.9 (0B71)—The Stacks project

Lemma 42.29.9. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X' \to X$ be a flat morphism of relative dimension $r$ of schemes locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1. Form the diagram

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

as in Remark 42.29.7. For any $(k + 1)$-cycle $\alpha $ on $X$ we have $(i')^*f^*\alpha = g^*i^*\alpha $ in $\mathop{\mathrm{CH}}\nolimits _{k + r}(D')$ (this makes sense as $f^*$ is defined on the level of cycles).

Proof. Suppose $\alpha = [W]$ for some integral closed subscheme $W \subset X$. Let $W' = f^{-1}(W) \subset X'$. In case $W \not\subset D$, then $W' \not\subset D'$ and we see that

\[ W' \cap D' = g^{-1}(W \cap D) \]

as closed subschemes of $D'$. Hence the equality holds as cycles, see Lemma 42.14.4. In case $W \subset D$, then $W' \subset D'$ and $W' = g^{-1}(W)$ with $[W']_{k + 1 + r} = g^*[W]$ and equality holds in $\mathop{\mathrm{CH}}\nolimits _{k + r}(D')$ by Lemma 42.26.2. By Remark 42.19.6 the result follows for general $\alpha '$. $\square$

Comments (4)

Comment #6638 by WhatJiaranEatsTonight on October 07, 2021 at 01:40

For any $(k+1)$ -cycle $\alpha$ on $X$ we have $(i')^*f^*\alpha=g^*i^*\alpha$ . The last $\alpha$ has an extra $'$ .

Comment #6863 by Johan on January 13, 2022 at 09:46

Thanks and fixed here.

Comment #7908 by qyk on December 06, 2022 at 11:11

Should the final $\mathrm{CH}_{k+r}(D)$ be $\mathrm{CH}_{k+r}(D')$ ?

Comment #8166 by Aise Johan de Jong on February 28, 2023 at 09:28

Thanks and fixed here.

Comments (4)

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