Lemma 42.63.3. Let $(S, \delta )$ be as above. Let $X$ be a scheme locally of finite type over $S$. Let $c \in A^ p(X \to S)$. Let $Y \to Z$ be a morphism of schemes locally of finite type over $S$. Let $c' \in A^ q(Y \to Z)$. Then $c \circ c' = c' \circ c$ in $A^{p + q}(X \times _ S Y \to X \times _ S Z)$.

Proof. In the proof of Lemma 42.63.2 we have seen that $c$ is given by a combination of proper pushforward, multiplying by integers over connected components, flat pullback, and gysin maps. Since $c'$ commutes with each of these operations by definition of bivariant classes, we conclude. Some details omitted. $\square$

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