The Stacks project

Lemma 42.24.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$, $\mathcal{N}$ be an invertible sheaves on $X$. Then

\[ c_1(\mathcal{L}) \cap \alpha + c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}) \cap \alpha \]

in $\mathop{\mathrm{CH}}\nolimits _ k(X)$ for every $\alpha \in Z_{k + 1}(X)$. Moreover, $c_1(\mathcal{O}_ X) \cap \alpha = 0$ for all $\alpha $.

Proof. The additivity follows directly from Divisors, Lemma 31.27.5 and the definitions. To see that $c_1(\mathcal{O}_ X) \cap \alpha = 0$ consider the section $1 \in \Gamma (X, \mathcal{O}_ X)$. This restricts to an everywhere nonzero section on any integral closed subscheme $W \subset X$. Hence $c_1(\mathcal{O}_ X) \cap [W] = 0$ as desired. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 42.24: Intersecting with an invertible sheaf

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Your email address will not be published.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 02SP. Beware of the difference between the letter 'O' and the digit '0'.