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
$\alpha \times \beta $ and $[\Delta ]$ intersect properly
we have $\Delta _*(\alpha \cdot \beta ) = [\Delta ] \cdot \alpha \times \beta $ as cycles on $X \times X$,
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
and the second by the integer
However, by Lemma 43.19.1 we have
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. $\square$
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