The Stacks project

Lemma 81.28.3. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The first Chern class of $\mathcal{L}$ on $X$ of Definition 81.28.2 is equal to the bivariant class of Lemma 81.26.4.

Proof. Namely, in this case $P = \mathbf{P}(\mathcal{L}) = X$ with $\mathcal{O}_ P(1) = \mathcal{L}$ by our normalization of the projective bundle, see Section 81.27. Hence the equation in Lemma 81.28.1 reads

\[ (-1)^0 c_1(\mathcal{L})^0 \cap c^{new}_1(\mathcal{L}) \cap \alpha + (-1)^1 c_1(\mathcal{L})^1 \cap c^{new}_0(\mathcal{L}) \cap \alpha = 0 \]

where $c_ i^{new}(\mathcal{L})$ is as in Definition 81.28.2. Since $c_0^{new}(\mathcal{L}) = 1$ and $c_1(\mathcal{L})^0 = 1$ we conclude. $\square$


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Your email address will not be published.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 0ES0. Beware of the difference between the letter 'O' and the digit '0'.