Lemma 42.47.10. In Lemma 42.47.1 assume $E_2$ has constant rank $0$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then

\[ c'_ i(E_2 \otimes \mathcal{L}) = \sum \nolimits _{j = 0}^ i \binom {- i + j}{j} c'_{i - j}(E_2) c_1(\mathcal{L})^ j \]

**Proof.**
The assumption on rank implies that $E_2|_{X_1 \cap X_2}$ is zero. Hence $c'_ i(E_2)$ is defined for all $i \geq 1$ and the statement makes sense. The actual equality follows immediately from Lemma 42.46.10 and the characterization of $c'_ i$ in Lemma 42.47.1.
$\square$

## 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Your email address will not be published.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.

## Comments (0)