The Stacks project

Lemma 42.37.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. If $\alpha \sim _{rat} \beta $ are rationally equivalent $k$-cycles on $X$ then $c_ j(\mathcal{E}) \cap \alpha = c_ j(\mathcal{E}) \cap \beta $ in $\mathop{\mathrm{CH}}\nolimits _{k - j}(X)$.

Proof. By Lemma 42.37.2 the elements $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, $j \geq 1$ and $\beta _ j = c_ j(\mathcal{E}) \cap \beta $, $j \geq 1$ are uniquely determined by the same equation in the chow group of the projective bundle associated to $\mathcal{E}$. (This of course relies on the fact that flat pullback is compatible with rational equivalence, see Lemma 42.20.2.) Hence they are equal. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 42.37: Intersecting with Chern classes

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Your email address will not be published.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 02U7. Beware of the difference between the letter 'O' and the digit '0'.