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 $Z_{r + s}(X \times Y)$.

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