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

## 43.21 Flat pullback and intersection products

Short discussion of the interaction between intersections and flat pullback.

**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

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 $f^*[W] = [f^{-1}(W)]_{s + e} = [f^*\mathcal{O}_ W]_{s + e}$.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)