Lemma 81.22.6. In Situation 81.2.1. Let $f : X' \to X$ be a flat morphism of relative dimension $r$ of good algebraic spaces over $B$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 81.22.1. Form the diagram

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

as in Remark 81.22.3. 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 subspace $W \subset X$. 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 subspaces of $D'$. Hence the equality holds as cycles, see Lemma 81.10.5. 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 81.19.2. By Remark 81.15.3 the result follows for general $\alpha '$. $\square$

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