The Stacks project

Lemma 42.67.5. In Situation 42.67.1 let $Y \to X \to S$ be locally of finite type and let $Y' \to X' \to S'$ be the base change by $S' \to S$. Assume $f : Y \to X$ is flat of relative dimension $r$. Then $f' : Y' \to X'$ is flat of relative dimension $r$ and the diagrams

\[ \vcenter { \xymatrix{ Z_{k + r}(Y) \ar[r]_{g^*} & Z_{k + c + r}(Y') \\ Z_ k(X) \ar[r]^{g^*} \ar[u]^{(f')^*} & Z_{k + c}(X') \ar[u]_{f^*} } } \quad \text{and}\quad \vcenter { \xymatrix{ \mathop{\mathrm{CH}}\nolimits _{k + r}(Y) \ar[r]_{g^*} & \mathop{\mathrm{CH}}\nolimits _{k + c + r}(Y') \\ \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r]^{g^*} \ar[u]^{(f')^*} & \mathop{\mathrm{CH}}\nolimits _{k + c}(X') \ar[u]_{f^*} } } \]

of cycle and chow groups commutes.

Proof. It suffices to show the first diagram commutes. To see this, let $Z \subset X$ be an integral closed subscheme of $\delta $-dimension $k$ and denote $Z' \subset X'$ its base change. By construction we have $g^*[Z] = [Z']_{k + c}$. By Lemma 42.14.4 we have $(f')^*g^*[Z] = [Z' \times _{X'} Y']_{k + c + r}$. Conversely, we have $f^*[Z] = [Z \times _ X Y]_{k + r}$ by Definition 42.14.1. By Lemma 42.67.3 we have $g^*f^*[Z] = [(Z \times _ X Y)']_{k + r + c}$. Since $(Z \times _ X Y)' = Z' \times _{X'} Y'$ by associativity of fibre product we conclude. $\square$


Comments (1)

Comment #9773 by gad on

For the diagrams to make sense, f^ and (f')^ should be interchanged.


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Your email address will not be published.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 0FVK. Beware of the difference between the letter 'O' and the digit '0'.