The Stacks project

Lemma 45.12.3. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then

\[ c^ A_ i({\mathcal E} \otimes {\mathcal L}) = \sum \nolimits _{j = 0}^ i \binom {r - i + j}{j} c^ A_{i - j}({\mathcal E}) \cup c^ A_1({\mathcal L})^ j \]

Proof. By the construction of $c^ A_ i$ we may assume $\mathcal{E}$ has constant rank $r$. Let $p : P \to X$ and $p' : P' \to X$ be the projective bundle associated to $\mathcal{E}$ and $\mathcal{E} \otimes \mathcal{L}$. Then there is an isomorphism $g : P \to P'$ such that $g^*\mathcal{O}_{P'}(1) = \mathcal{O}_ P(1) \otimes p^*\mathcal{L}$. See Constructions, Lemma 27.20.1. Thus

\[ g^*c_1^ A(\mathcal{O}_{P'}(1)) = c_1^ A(\mathcal{O}_ P(1)) + p^*c_1^ A(\mathcal{L}) \]

The desired equality follows formally from this and the definition of Chern classes using equation (45.12.1.1). $\square$

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Your email address will not be published.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 0FI8. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 45.12.3 as pdf