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 $\alpha \times \beta \sim _{rat} 0$.
43.18 Exterior product
Let $X$ and $Y$ be varieties. Let $V$, resp. $W$ be a closed subvariety of $X$, resp. $Y$. The product $V\times W$ is a closed subvariety of $X\times Y$ (Lemma 43.13.1). For a $k$-cycle $\alpha = \sum n_ i [V_ i]$ and a $l$-cycle $\beta = \sum m_ j [V_ j]$ on $Y$ we define the exterior product of $\alpha $ and $\beta $ to be the cycle $\alpha \times \beta = \sum n_ i m_ j [V_ i \times W_ j]$. Exterior product defines a $\mathbf{Z}$-linear map
\[ Z_ r(X) \otimes _\mathbf {Z} Z_ s(Y) \longrightarrow Z_{r + s}(X \times Y) \]
Let us prove that exterior product factors through rational equivalence.
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 $\kappa (\xi )$. $\square$
We conclude that exterior product defines a commutative diagram
\[ \xymatrix{ Z_ r(X) \otimes _\mathbf {Z} Z_ s(Y) \ar[r] \ar[d] & Z_{r + s}(X \times Y) \ar[d] \\ \mathop{\mathrm{CH}}\nolimits _ r(X) \otimes _\mathbf {Z} \mathop{\mathrm{CH}}\nolimits _ s(Y) \ar[r] & \mathop{\mathrm{CH}}\nolimits _{r + s}(X \times Y) } \]
for any pair of varieties $X$ and $Y$. For nonsingular varieties we can think of the exterior product as an intersection product of pullbacks.
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 $Z_{r + s}(X \times Y)$.
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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