See [Chapter V, C), Section 7, formula (10), Serre_algebre_locale] for a more general formula.

Lemma 43.22.1. Let $f : X \to Y$ be a flat proper morphism of nonsingular varieties. Set $e = \dim (X) - \dim (Y)$. Let $\alpha$ be an $r$-cycle on $X$ and let $\beta$ be a $s$-cycle on $Y$. Assume that $\alpha$ and $f^*(\beta )$ intersect properly. Then $f_*(\alpha )$ and $\beta$ intersect properly and

$f_*(\alpha ) \cdot \beta = f_*( \alpha \cdot f^*\beta )$

Proof. By linearity we reduce to the case where $\alpha = [V]$ and $\beta = [W]$ for some closed subvariety $V \subset X$ and $W \subset Y$ of dimension $r$ and $s$. Then $f^{-1}(W)$ has pure dimension $s + e$. We assume the cycles $[V]$ and $f^*[W]$ intersect properly. We will use without further mention the fact that $V \cap f^{-1}(W) \to f(V) \cap W$ is surjective.

Let $a$ be the dimension of the generic fibre of $V \to f(V)$. If $a > 0$, then $f_*[V] = 0$. In particular $f_*\alpha$ and $\beta$ intersect properly. To finish this case we have to show that $f_*([V] \cdot f^*[W]) = 0$. However, since every fibre of $V \to f(V)$ has dimension $\geq a$ (see Morphisms, Lemma 29.28.4) we conclude that every irreducible component $Z$ of $V \cap f^{-1}(W)$ has fibres of dimension $\geq a$ over $f(Z)$. This certainly implies what we want.

Assume that $V \to f(V)$ is generically finite. Let $Z \subset f(V) \cap W$ be an irreducible component. Let $Z_ i \subset V \cap f^{-1}(W)$, $i = 1, \ldots , t$ be the irreducible components of $V \cap f^{-1}(W)$ dominating $Z$. By assumption each $Z_ i$ has dimension $r + s + e - \dim (X) = r + s - \dim (Y)$. Hence $\dim (Z) \leq r + s - \dim (Y)$. Thus we see that $f(V)$ and $W$ intersect properly, $\dim (Z) = r + s - \dim (Y)$, and each $Z_ i \to Z$ is generically finite. In particular, it follows that $V \to f(V)$ has finite fibre over the generic point $\xi$ of $Z$. Thus $V \to Y$ is finite in an open neighbourhood of $\xi$, see Cohomology of Schemes, Lemma 30.21.2. Using a very general projection formula for derived tensor products, we get

$Rf_*(\mathcal{O}_ V \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{O}_ W) = Rf_*\mathcal{O}_ V \otimes _{\mathcal{O}_ Y}^\mathbf {L} \mathcal{O}_ W$

see Derived Categories of Schemes, Lemma 36.22.1. Since $f$ is flat, we see that $Lf^*\mathcal{O}_ W = f^*\mathcal{O}_ W$. Since $f|_ V$ is finite in an open neighbourhood of $\xi$ we have

$(Rf_*\mathcal{F})_\xi = (f_*\mathcal{F})_\xi$

for any coherent sheaf on $X$ whose support is contained in $V$ (see Cohomology of Schemes, Lemma 30.20.8). Thus we conclude that

43.22.1.1
\begin{equation} \label{intersection-equation-stalks} \left( f_*\text{Tor}_ i^{\mathcal{O}_ X}(\mathcal{O}_ V, f^*\mathcal{O}_ W) \right)_\xi = \left(\text{Tor}_ i^{\mathcal{O}_ Y}(f_*\mathcal{O}_ V, \mathcal{O}_ W)\right)_\xi \end{equation}

for all $i$. Since $f^*[W] = [f^*\mathcal{O}_ W]_{s + e}$ by Lemma 43.7.1 we have

$[V] \cdot f^*[W] = \sum (-1)^ i [\text{Tor}_ i^{\mathcal{O}_ X}(\mathcal{O}_ V, f^*\mathcal{O}_ W)]_{r + s - \dim (Y)}$

by Lemma 43.19.4. Applying Lemma 43.6.1 we find

$f_*([V] \cdot f^*[W]) = \sum (-1)^ i [f_*\text{Tor}_ i^{\mathcal{O}_ X}(\mathcal{O}_ V, f^*\mathcal{O}_ W)]_{r + s - \dim (Y)}$

Since $f_*[V] = [f_*\mathcal{O}_ V]_ r$ by Lemma 43.6.1 we have

$[f_*V] \cdot [W] = \sum (-1)^ i [\text{Tor}_ i^{\mathcal{O}_ X}(f_*\mathcal{O}_ V, \mathcal{O}_ W)]_{r + s - \dim (Y)}$

again by Lemma 43.19.4. Comparing the formula for $f_*([V] \cdot f^*[W])$ with the formula for $f_*[V] \cdot [W]$ and looking at the coefficient of $Z$ by taking lengths of stalks at $\xi$, we see that (43.22.1.1) finishes the proof. $\square$

Comment #5535 by on

The reference is not precise. I think it is better to write [Chapter V, C), Section 7, formula (10), Serre_algebre_locale]

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