The Stacks project

Lemma 62.11.5. Let $(S, \delta )$ and $f : X \to Y$ be as above. Let $Z \subset X$ be a closed subscheme of relative dimension $\leq r$ over $Y$. Set $\alpha = [Z/X/Y]_ r$ (Example 62.5.4). Let $W \subset Y$ be a closed subscheme of $\delta $-dimension $\leq e$. Set $\beta = [W]_ e$ (Chow Homology, Definition 42.9.2). If $Z$ is flat over $Y$, then $\alpha \cap \beta = [Z \times _ Y W]_{r + e}$.

Proof. This is a special case of Lemma 62.11.4 if we take $\mathcal{F} = \mathcal{O}_ Z$ and $\mathcal{F} = \mathcal{O}_ W$. $\square$

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