The Stacks project

Proof. By linearity and symmetry in $X$ and $Y$, it suffices to prove this when $\alpha = [V]$ for some subvariety $V \subset X$ of dimension $r$ and $\beta = [W_ a]_ s - [W_ b]_ s$ for some closed subvariety $W \subset Y \times \mathbf{P}^1$ of dimension $s + 1$ which intersects $Y \times a$ and $Y \times b$ properly. In this case the lemma follows if we can prove

and similarly with $a$ replaced by $b$. Namely, then we see that $\alpha \times \beta = [(V \times W)_ a]_{r + s} - [(V \times W)_ b]_{r + s}$ as desired. To see the displayed equality we note the equality

of schemes. The projection $V \times W_ a \to W_ a$ induces a bijection of irreducible components (see for example Varieties, Lemma 33.8.4). Let $W' \subset W_ a$ be an irreducible component with generic point $\zeta $. Then $V \times W'$ is the corresponding irreducible component of $V \times W_ a$ (see Lemma 43.13.1). Let $\xi $ be the generic point of $V \times W'$. We have to show that

In this formula we may replace $\mathcal{O}_{Y, \zeta }$ by $\mathcal{O}_{W_ a, \zeta }$ and we may replace $\mathcal{O}_{X \times Y, \zeta }$ by $\mathcal{O}_{V \times W_ a, \zeta }$ (see Algebra, Lemma 10.52.5). As $\mathcal{O}_{W_ a, \zeta } \to \mathcal{O}_{V \times W_ a, \xi }$ is flat, by Algebra, Lemma 10.52.13 it suffices to show that

This is true because the quotient on the right is the local ring $\mathcal{O}_{V \times W', \xi }$ of a variety at a generic point hence equal to $\kappa (\xi )$. $\square$