Lemma 42.34.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Assume we have disjoint union decompositions $X = \coprod _{i \in I} X_ i$ and $Y = \coprod _{j \in J} Y_ j$ by open and closed subschemes and a map $a : I \to J$ of sets such that $f(X_ i) \subset Y_{a(i)}$. Then

$A^ p(X \to Y) = \prod \nolimits _{i \in I} A^ p(X_ i \to Y_{a(i)})$

Proof. Suppose given an element $(c_ i) \in \prod _ i A^ p(X_ i \to Y_{a(i)})$. Then given $\beta \in \mathop{\mathrm{CH}}\nolimits _ k(Y)$ we can map this to the element of $\mathop{\mathrm{CH}}\nolimits _{k - p}(X)$ whose restriction to $X_ i$ is $c_ i \cap \beta |_{Y_{a(i)}}$. This works because $\mathop{\mathrm{CH}}\nolimits _{k - p}(X) = \prod _ i \mathop{\mathrm{CH}}\nolimits _{k - p}(X_ i)$. The same construction works after base change by any $Y' \to Y$ locally of finite type and we get $c \in A^ p(X \to Y)$. Thus we obtain a map $\Psi$ from the right hand side of the formula to the left hand side of the formula. Conversely, given $c \in A^ p(X \to Y)$ and an element $\beta _ i \in \mathop{\mathrm{CH}}\nolimits _ k(Y_{a(i)})$ we can consider the element $(c \cap (Y_{a(i)} \to Y)_*\beta _ i)|_{X_ i}$ in $\mathop{\mathrm{CH}}\nolimits _{k - p}(X_ i)$. The same thing works after base change by any $Y' \to Y$ locally of finite type and we get $c_ i \in A^ p(X_ i \to Y_{a(i)})$. Thus we obtain a map $\Phi$ from the left hand side of the formula to the right hand side of the formula. It is immediate that $\Phi \circ \Psi = \text{id}$. For the converse, suppose that $c \in A^ p(X \to Y)$ and $\beta \in \mathop{\mathrm{CH}}\nolimits _ k(Y)$. Say $\Phi (c) = (c_ i)$. Let $j \in J$. Because $c$ commutes with flat pullback we get

$(c \cap \beta )|_{\coprod _{a(i) = j} X_ i} = c \cap \beta |_{Y_ j}$

Because $c$ commutes with proper pushforward we get

$(\coprod \nolimits _{a(i) = j} X_ i \to X)_* ((c \cap \beta )|_{\coprod _{a(i) = j} X_ i}) = c \cap (Y_ j \to Y)_*\beta |_{Y_ j}$

The left hand side is the cycle on $X$ restricting to $(c \cap \beta )|_{X_ i}$ on $X_ i$ for $i \in I$ with $a(i) = j$ and $0$ else. The right hand side is a cycle on $X$ whose restriction to $X_ i$ is $c_ i \cap \beta |_{Y_ j}$ for $i \in I$ with $a(i) = j$. Thus $c \cap \beta = \Psi ((c_ i))$ as desired. $\square$

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