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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