43.21 Flat pullback and intersection products
Short discussion of the interaction between intersections and flat pullback.
Lemma 43.21.1. Let $f : X \to Y$ be a flat morphism of nonsingular varieties. Set $e = \dim (X) - \dim (Y)$. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $Y$ with $\dim (\text{Supp}(\mathcal{F})) \leq r$, $\dim (\text{Supp}(\mathcal{G})) \leq s$, and $\dim (\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G}) ) \leq r + s - \dim (Y)$. In this case the cycles $[f^*\mathcal{F}]_{r + e}$ and $[f^*\mathcal{G}]_{s + e}$ intersect properly and
\[ f^*([\mathcal{F}]_ r \cdot [\mathcal{G}]_ s) = [f^*\mathcal{F}]_{r + e} \cdot [f^*\mathcal{G}]_{s + e} \]
Proof.
The statement that $[f^*\mathcal{F}]_{r + e}$ and $[f^*\mathcal{G}]_{s + e}$ intersect properly is immediate from the assumption that $f$ has relative dimension $e$. By Lemmas 43.19.4 and 43.7.1 it suffices to show that
\[ f^*\text{Tor}_ i^{\mathcal{O}_ Y}(\mathcal{F}, \mathcal{G}) = \text{Tor}_ i^{\mathcal{O}_ X}(f^*\mathcal{F}, f^*\mathcal{G}) \]
as $\mathcal{O}_ X$-modules. This follows from Cohomology, Lemma 20.27.3 and the fact that $f^*$ is exact, so $Lf^*\mathcal{F} = f^*\mathcal{F}$ and similarly for $\mathcal{G}$.
$\square$
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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