The Stacks project

43.22 Projection formula for flat proper morphisms

Short discussion of the projection formula for flat proper morphisms.


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
\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 ( finishes the proof. $\square$

Lemma 43.22.2. Let $X \to P$ be a closed immersion of nonsingular varieties. Let $C' \subset P \times \mathbf{P}^1$ be a closed subvariety of dimension $r + 1$. Assume

  1. the fibre $C = C'_0$ has dimension $r$, i.e., $C' \to \mathbf{P}^1$ is dominant,

  2. $C'$ intersects $X \times \mathbf{P}^1$ properly,

  3. $[C]_ r$ intersects $X$ properly.

Then setting $\alpha = [C]_ r \cdot X$ viewed as cycle on $X$ and $\beta = C' \cdot X \times \mathbf{P}^1$ viewed as cycle on $X \times \mathbf{P}^1$, we have

\[ \alpha = \text{pr}_{X, *}(\beta \cdot X \times 0) \]

as cycles on $X$ where $\text{pr}_ X : X \times \mathbf{P}^1 \to X$ is the projection.

Proof. Let $\text{pr} : P \times \mathbf{P}^1 \to P$ be the projection. Since we are proving an equality of cycles it suffices to think of $\alpha $, resp. $\beta $ as a cycle on $P$, resp. $P \times \mathbf{P}^1$ and prove the result for pushing forward by $\text{pr}$. Because $\text{pr}^*X = X \times \mathbf{P}^1$ and $\text{pr}$ defines an isomorphism of $C'_0$ onto $C$ the projection formula (Lemma 43.22.1) gives

\[ \text{pr}_*([C'_0]_ r \cdot X \times \mathbf{P}^1) = [C]_ r \cdot X = \alpha \]

On the other hand, we have $[C'_0]_ r = C' \cdot P \times 0$ as cycles on $P \times \mathbf{P}^1$ by Lemma 43.17.1. Hence

\[ [C'_0]_ r \cdot X \times \mathbf{P}^1 = (C' \cdot P \times 0) \cdot X \times \mathbf{P}^1 = (C' \cdot X \times \mathbf{P}^1) \cdot P \times 0 \]

by associativity (Lemma 43.20.1) and commutativity of the intersection product. It remains to show that the intersection product of $C' \cdot X \times \mathbf{P}^1$ with $P \times 0$ on $P \times \mathbf{P}^1$ is equal (as a cycle) to the intersection product of $\beta $ with $X \times 0$ on $X \times \mathbf{P}^1$. Write $C' \cdot X \times \mathbf{P}^1 = \sum n_ k[E_ k]$ and hence $\beta = \sum n_ k[E_ k]$ for some subvarieties $E_ k \subset X \times \mathbf{P}^1 \subset P \times \mathbf{P}^1$. Then both intersections are equal to $\sum m_ k[E_{k, 0}]$ by Lemma 43.17.1 applied twice. This finishes the proof. $\square$

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