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 $s$ 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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (1)
Comment #8665 by KrLee on