The Stacks project

Lemma 43.19.2. Let $X$ be a nonsingular variety. Let $\alpha $, resp. $\beta $ be an $r$-cycle, resp. $s$-cycle on $X$. Assume $\alpha $ and $\beta $ intersect properly. Then

  1. $\alpha \times \beta $ and $[\Delta ]$ intersect properly

  2. we have $\Delta _*(\alpha \cdot \beta ) = [\Delta ] \cdot \alpha \times \beta $ as cycles on $X \times X$,

  3. if $X$ is proper, then $\text{pr}_{1, *}([\Delta ] \cdot \alpha \times \beta ) = \alpha \cdot \beta $, where $pr_1 : X\times X \to X$ is the projection.

Proof. By linearity it suffices to prove this when $\alpha = [V]$ and $\beta = [W]$ for some closed subvarieties $V \subset X$ and $W \subset Y$ which intersect properly. Recall that $V \times W$ is a closed subvariety of dimension $r + s$. Observe that scheme theoretically we have $V \cap W = \Delta ^{-1}(V \times W)$ as well as $\Delta (V \cap W) = \Delta \cap V \times W$. This proves (1).

Proof of (2). Let $Z \subset V \cap W$ be an irreducible component with generic point $\xi $. We have to show that the coefficient of $Z$ in $\alpha \cdot \beta $ is the same as the coefficient of $\Delta (Z)$ in $[\Delta ] \cdot \alpha \times \beta $. The first is given by the integer

\[ \sum (-1)^ i \text{length}_{\mathcal{O}_{X, \xi }} \text{Tor}_ i^{\mathcal{O}_ X}(\mathcal{O}_ V, \mathcal{O}_ W)_\xi \]

and the second by the integer

\[ \sum (-1)^ i \text{length}_{\mathcal{O}_{X \times Y, \Delta (\xi )}} \text{Tor}_ i^{\mathcal{O}_{X \times Y}}( \mathcal{O}_\Delta , \mathcal{O}_{V \times W})_{\Delta (\xi )} \]

However, by Lemma 43.19.1 we have

\[ \text{Tor}_ i^{\mathcal{O}_ X}(\mathcal{O}_ V, \mathcal{O}_ W)_\xi \cong \text{Tor}_ i^{\mathcal{O}_{X \times Y}}( \mathcal{O}_\Delta , \mathcal{O}_{V \times W})_{\Delta (\xi )} \]

as $\mathcal{O}_{X \times X, \Delta (\xi )}$-modules. Thus equality of lengths (by Algebra, Lemma 10.52.5 to be precise).

Part (2) implies (3) because $\text{pr}_{1, *} \circ \Delta _* = \text{id}$ by Lemma 43.6.2. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B0U. Beware of the difference between the letter 'O' and the digit '0'.