The Stacks project

Remark 59.22.2. In the statement of Lemma 59.22.1 the covering $\mathcal{U}$ is a refinement of $\mathcal{V}$ but not the other way around. Coverings of the form $\{ V \to U\} $ do not form an initial subcategory of the category of all coverings of $U$. Yet it is still true that we can compute Čech cohomology $\check H^ n(U, \mathcal{F})$ (which is defined as the colimit over the opposite of the category of coverings $\mathcal{U}$ of $U$ of the Čech cohomology groups of $\mathcal{F}$ with respect to $\mathcal{U}$) in terms of the coverings $\{ V \to U\} $. We will formulate a precise lemma (it only works for sheaves) and add it here if we ever need it.

Comments (0)

There are also:

  • 3 comment(s) on Section 59.22: Cohomology of quasi-coherent sheaves

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Your email address will not be published.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 03P0. Beware of the difference between the letter 'O' and the digit '0'.