The Stacks project

59.5 Feats of the étale topology

For a natural number $n \in \mathbf{N} = \{ 1, 2, 3, 4, \ldots \} $ it is true that

\[ H_{\acute{e}tale}^2 (\mathbf{P}^1_\mathbf {C}, \mathbf{Z}/n\mathbf{Z}) = \mathbf{Z}/n\mathbf{Z}. \]

More generally, if $X$ is a complex variety, then its étale Betti numbers with coefficients in a finite field agree with the usual Betti numbers of $X(\mathbf{C})$, i.e.,

\[ \dim _{\mathbf{F}_ q} H_{\acute{e}tale}^{2i} (X, \mathbf{F}_ q) = \dim _{\mathbf{F}_ q} H_{Betti}^{2i} (X(\mathbf{C}), \mathbf{F}_ q). \]

This is extremely satisfactory. However, these equalities only hold for torsion coefficients, not in general. For integer coefficients, one has

\[ H_{\acute{e}tale}^2 (\mathbf{P}^1_\mathbf {C}, \mathbf{Z}) = 0. \]

By contrast $H_{Betti}^2(\mathbf{P}^1(\mathbf{C}), \mathbf{Z}) = \mathbf{Z}$ as the topological space $\mathbf{P}^1(\mathbf{C})$ is homeomorphic to a $2$-sphere. There are ways to get back to nontorsion coefficients from torsion ones by a limit procedure which we will come to shortly.

Comments (2)

Comment #1700 by Yogesh More on

Very minor remark: I think it would be helpful to add, after the sentence "For integer coefficients, one has ", the following:

By contrast since is topologically equivalent to .

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Your email address will not be published.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 03N7. Beware of the difference between the letter 'O' and the digit '0'.