The Stacks project

Lemma 43.18.1. Let $X$ and $Y$ be varieties. Let $\alpha \in Z_ r(X)$ and $\beta \in Z_ s(Y)$. If $\alpha \sim _{rat} 0$ or $\beta \sim _{rat} 0$, then $\alpha \times \beta \sim _{rat} 0$.

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

\[ [(V \times W)_ a]_{r + s} = [V] \times [W_ a]_ s \]

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

\[ V \times W_ a = (V \times W)_ a \]

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

\[ \text{length}_{\mathcal{O}_{Y, \zeta }}(\mathcal{O}_{W_ a, \zeta }) = \text{length}_{\mathcal{O}_{X \times Y, \xi }}( \mathcal{O}_{V \times W_ a, \xi }) \]

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

\[ \text{length}_{\mathcal{O}_{V \times W_ a, \xi }}( \mathcal{O}_{V \times W_ a, \xi }/ \mathfrak m_\zeta \mathcal{O}_{V \times W_ a, \xi }) = 1 \]

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 $\kappa (\xi )$. $\square$

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