Lemma 43.18.2. Let $X$ and $Y$ be nonsingular varieties. Let $\alpha \in Z_ r(X)$ and $\beta \in Z_ s(Y)$. Then
$\text{pr}_ Y^*(\beta ) = [X] \times \beta $ and $\text{pr}_ X^*(\alpha ) = \alpha \times [Y]$,
$\alpha \times [Y]$ and $[X]\times \beta $ intersect properly on $X\times Y$, and
we have $\alpha \times \beta = (\alpha \times [Y])\cdot ([X]\times \beta ) = pr_ Y^*(\alpha ) \cdot pr_ X^*(\beta )$ in $Z_{r + s}(X \times Y)$.
Proof. By linearity we may assume $\alpha = [V]$ and $\beta = [W]$. Then (1) says that $\text{pr}_ Y^{-1}(W) = X \times W$ and $\text{pr}_ X^{-1}(V) = V \times Y$. This is clear. Part (2) holds because $X \times W \cap V \times Y = V \times W$ and $\dim (V \times W) = r + s$ by Lemma 43.13.1.
Proof of (3). Let $\xi $ be the generic point of $V \times W$. Since the projections $X \times W \to W$ is smooth as a base change of $X \to \mathop{\mathrm{Spec}}(\mathbf{C})$, we see that $X \times W$ is nonsingular at every point lying over the generic point of $W$, in particular at $\xi $. Similarly for $V \times Y$. Hence $\mathcal{O}_{X \times W, \xi }$ and $\mathcal{O}_{V \times Y, \xi }$ are Cohen-Macaulay local rings and Lemma 43.16.1 applies. Since $V \times Y \cap X \times W = V \times W$ scheme theoretically the proof is complete. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: