The Stacks project

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 $\mathcal{G}$. $\square$

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