The Stacks project

Lemma 43.18.2. Let $X$ and $Y$ be nonsingular varieties. Let $\alpha \in Z_ r(X)$ and $\beta \in Z_ s(Y)$. Then

  1. $\text{pr}_ Y^*(\beta ) = [X] \times \beta $ and $\text{pr}_ X^*(\alpha ) = \alpha \times [Y]$,

  2. $\alpha \times [Y]$ and $[X]\times \beta $ intersect properly on $X\times Y$, and

  3. 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. $\square$

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