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
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
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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