The Stacks project

Lemma 42.62.7. Let $k$ be a field. Let $X$ be a smooth scheme over $k$ which is quasi-compact and has affine diagonal. Then the intersection product on $\mathop{\mathrm{CH}}\nolimits ^*(X)$ constructed in this section agrees after tensoring with $\mathbf{Q}$ with the intersection product constructed in Section 42.58.

Proof. Let $\alpha \in \mathop{\mathrm{CH}}\nolimits ^ i(X)$ and $\beta \in \mathop{\mathrm{CH}}\nolimits ^ j(X)$. Write $\alpha = ch(\alpha ') \cap [X]$ and $\beta = ch(\beta ') \cap [X]$ $\alpha ', \beta ' \in K_0(\textit{Vect}(X)) \otimes \mathbf{Q}$ as in Section 42.58. Set $c = ch(\alpha ')$ and $c' = ch(\beta ')$. Then the intersection product in Section 42.58 produces $c \cap c' \cap [X]$. This is the same as $\alpha \cdot \beta $ by Lemma 42.62.2 (or rather the generalization that $A^ i(X) \to \mathop{\mathrm{CH}}\nolimits ^ i(X)$, $c \mapsto c \cap [X]$ is an isomorphism for any smooth scheme $X$ over $k$). $\square$

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