The Stacks project

Lemma 42.47.7. In Lemma 42.47.1 assume $E_2|_{X_1 \cap X_2}$ is zero. Then

\begin{align*} P'_1(E_2) & = c'_1(E_2), \\ P'_2(E_2) & = c'_1(E_2)^2 - 2c'_2(E_2), \\ P'_3(E_2) & = c'_1(E_2)^3 - 3c'_1(E_2)c'_2(E_2) + 3c'_3(E_2), \\ P'_4(E_2) & = c'_1(E_2)^4 - 4c'_1(E_2)^2c'_2(E_2) + 4c'_1(E_2)c'_3(E_2) + 2c'_2(E_2)^2 - 4c'_4(E_2), \end{align*}

and so on with multiplication as in Remark 42.34.7.

Proof. The statement makes sense because the zero sheaf has rank $< 1$ and hence the classes $c'_ p(E_2)$ are defined for all $p \geq 1$. The equalities follow immediately from the characterization of the classes produced by Lemma 42.47.1 and the corresponding result for capping with the Chern classes of $E_2$ given in Remark 42.46.8. $\square$

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Your email address will not be published.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



View Lemma 42.47.7 as pdf