The Stacks project

Lemma 43.21.2. Let $f : X \to Y$ be a flat morphism of nonsingular varieties. Let $\alpha $ be a $r$-cycle on $Y$ and $\beta $ an $s$-cycle on $Y$. Assume that $\alpha $ and $\beta $ intersect properly. Then $f^*\alpha $ and $f^*\beta $ intersect properly and $f^*( \alpha \cdot \beta ) = f^*\alpha \cdot f^*\beta $.

Proof. By linearity we may assume that $\alpha = [V]$ and $\beta = [W]$ for some closed subvarieties $V, W \subset Y$ of dimension $r, s$. Say $f$ has relative dimension $e$. Then the lemma is a special case of Lemma 43.21.1 because $[V] = [\mathcal{O}_ V]_ r$, $[W] = [\mathcal{O}_ W]_ r$, $f^*[V] = [f^{-1}(V)]_{r + e} = [f^*\mathcal{O}_ V]_{r + e}$, and $f^*[W] = [f^{-1}(W)]_{s + e} = [f^*\mathcal{O}_ W]_{s + e}$. $\square$

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