The Stacks project

Lemma 42.67.8. In Situation 42.67.1 let $X \to S$ be locally of finite type and let $X' \to S'$ be the base change by $S' \to S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module of rank $r$ with base change $\mathcal{E}'$ on $X'$. Then the diagram

\[ \xymatrix{ \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r]_{g^*} \ar[d]_{c_ i(\mathcal{E}) \cap -} & \mathop{\mathrm{CH}}\nolimits _{k + c}(X') \ar[d]^{c_ i(\mathcal{E}') \cap -} \\ \mathop{\mathrm{CH}}\nolimits _{k - i}(X) \ar[r]^{g^*} & \mathop{\mathrm{CH}}\nolimits _{k + c - i}(X') } \]

of chow groups commutes for all $i$.

Proof. Set $P = \mathbf{P}(\mathcal{E})$. The base change $P'$ of $P$ is equal to $\mathbf{P}(\mathcal{E}')$. Since we already know that flat pullback and cupping with $c_1$ of an invertible module commute with base change (Lemmas 42.67.5 and 42.67.7) the lemma follows from the characterization of capping with $c_ i(\mathcal{E})$ given in Lemma 42.38.2. $\square$

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Your email address will not be published.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



View Lemma 42.67.8 as pdf