Lemma 42.67.9. Let $(S, \delta )$, $(S', \delta ')$, $(S'', \delta '')$ be as in Situation 42.7.1. Let $g : S' \to S$ and $g' : S'' \to S'$ be flat morphisms of schemes and let $c, c' \in \mathbf{Z}$ be integers such that $S, \delta , S', \delta ', g, c$ and $S', \delta ', S'', g', c'$ are as in Situation 42.67.1. Let $X \to S$ be locally of finite type and denote $X' \to S'$ and $X'' \to S''$ the base changes by $S' \to S$ and $S'' \to S$. Then

$S, \delta , S'', \delta '', g \circ g', c + c'$ is as in Situation 42.67.1,

the maps $g^* : Z_ k(X) \to Z_{k + c}(X')$ and $(g')^* : Z_{k + c}(X') \to Z_{k + c + c'}(X'')$ of compose to give the map $(g \circ g')^* : Z_ k(X) \to Z_{k + c + c'}(X'')$, and

the maps $g^* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k + c}(X')$ and $(g')^* : \mathop{\mathrm{CH}}\nolimits _{k + c}(X') \to \mathop{\mathrm{CH}}\nolimits _{k + c + c'}(X'')$ of Lemma 42.67.4 compose to give the map $(g \circ g')^* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k + c + c'}(X'')$ of Lemma 42.67.4.

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