The Stacks project

Proposition 62.12.3. Let $(S, \delta )$ be as in Section 62.11. Let $X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $r \geq 0$ and let $\alpha \in z(X/Y, r)$ be a relative $r$-cycle on $X/Y$. The rule that to every morphism $g : Y' \to Y$ locally of finite type and every $e \in \mathbf{Z}$ associates the operation

\[ g^*\alpha \cap - : Z_ e(Y') \to Z_{r + e}(X') \]

where $X' = Y' \times _ Y X$ factors through rational equivalence to define a bivariant class $c(\alpha ) \in A^{-r}(X \to Y)$.

Proof. The operation factors through rational equivalence by Lemma 62.12.2 and Chow Homology, Lemma 42.35.1. The resulting operation on chow groups is a bivariant class by Chow Homology, Lemma 42.35.2 and Lemmas 62.11.6, 62.11.3, and 62.12.2. $\square$

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