The Stacks project

Remark 42.34.8. Let $(S, \delta )$ be as in Situation 42.7.1. Let $Z \to X$ be a closed immersion of schemes locally of finite type over $S$. Denote $res : A^ p(Z \to X) \to A^ p(Z)$ the restriction map of Remark 42.33.5. For $c \in A^ p(Z \to X)$ we have $res(c) \cap \alpha = c \cap i_*\alpha $ for $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(Z)$. Namely $res(c) \cap \alpha = c \cap \alpha $ and compatibility of $c$ with proper pushforward gives $(Z \to Z)_*(c \cap \alpha ) = c \cap (Z \to X)_*\alpha $.