The Stacks project

Lemma 42.61.1. The map $\times : \mathop{\mathrm{CH}}\nolimits _ n(X) \otimes _{\mathbf{Z}} \mathop{\mathrm{CH}}\nolimits _ m(Y) \to \mathop{\mathrm{CH}}\nolimits _{n + m}(X \times _ k Y)$ is well defined.

Proof. A first remark is that if $\alpha = \sum n_ i[X_ i]$ and $\beta = \sum m_ j[Y_ j]$ with $X_ i \subset X$ and $Y_ j \subset Y$ locally finite families of integral closed subschemes of dimensions $n$ and $m$, then $X_ i \times _ k Y_ j$ is a locally finite collection of closed subschemes of $X \times _ k Y$ of dimensions $n + m$ and we can indeed consider

\[ \alpha \times \beta = \sum n_ i m_ j [X_ i \times _ k Y_ j]_{n + m} \]

as a $(n + m)$-cycle on $X \times _ k Y$. In this way we obtain an additive map $\times : Z_ n(X) \otimes _{\mathbf{Z}} Z_ m(Y) \to Z_{n + m}(X \times _ k Y)$. The problem is to show that this procedure is compatible with rational equivalence.

Let $i : X' \to X$ be the inclusion morphism of an integral closed subscheme of dimension $n$. Then flat pullback along the morphism $p' : X' \to \mathop{\mathrm{Spec}}(k)$ is an element $(p')^* \in A^{-n}(X' \to \mathop{\mathrm{Spec}}(k))$ by Lemma 42.33.2 and hence $c' = i_* \circ (p')^* \in A^{-n}(X \to \mathop{\mathrm{Spec}}(k))$ by Lemma 42.33.4. This produces maps

\[ c' \cap - : \mathop{\mathrm{CH}}\nolimits _ m(Y) \longrightarrow \mathop{\mathrm{CH}}\nolimits _{m + n}(X \times _ k Y) \]

which the reader easily sends $[Y']$ to $[X' \times _ k Y']_{n + m}$ for any integral closed subscheme $Y' \subset Y$ of dimension $m$. Hence the construction $([X'], [Y']) \mapsto [X' \times _ k Y']_{n + m}$ factors through rational equivalence in the second variable, i.e., gives a well defined map $Z_ n(X) \otimes _{\mathbf{Z}} \mathop{\mathrm{CH}}\nolimits _ m(Y) \to \mathop{\mathrm{CH}}\nolimits _{n + m}(X \times _ k Y)$. By symmetry the same is true for the other variable and we conclude. $\square$

Comments (0)

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FBV. Beware of the difference between the letter 'O' and the digit '0'.