The Stacks project

Lemma 42.47.9. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $i_ j : X_ j \to X$, $j = 1, 2$ be closed immersions such that $X = X_1 \cup X_2$ set theoretically. Let $E, F \in D(\mathcal{O}_{X_2})$ be perfect objects. Assume

  1. Chern classes of $E$ and $F$ are defined,

  2. the restrictions $E|_{X_1 \cap X_2}$ and $F|_{X_1 \cap X_2}$ are zero,

Denote $P'_ p(E), P'_ p(F), P'_ p(E \oplus F) \in A^ p(X_2 \to X)$ for $p \geq 0$ the classes constructed in Lemma 42.47.1. Then $P'_ p(E \oplus F) = P'_ p(E) + P'_ p(F)$.

Proof. Immediate from the characterization of the classes in Lemma 42.47.1 and the additivity in Lemma 42.46.7. $\square$

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Your email address will not be published.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



View Lemma 42.47.9 as pdf