The Stacks project

Lemma 21.10.6. Let $\mathcal{C}$ be a site. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. For any abelian sheaf $\mathcal{F}$ there is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with

\[ E_2^{p, q} = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F})) \]

converging to $H^{p + q}(U, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$.

Proof. This is a Grothendieck spectral sequence (see Derived Categories, Lemma 13.22.2) for the functors

\[ i : \textit{Ab}(\mathcal{C}) \to \textit{PAb}(\mathcal{C}) \quad \text{and}\quad \check{H}^0(\mathcal{U}, - ) : \textit{PAb}(\mathcal{C}) \to \textit{Ab}. \]

Namely, we have $\check{H}^0(\mathcal{U}, i(\mathcal{F})) = \mathcal{F}(U)$ by Lemma 21.8.2. We have that $i(\mathcal{I})$ is Čech acyclic by Lemma 21.10.2. And we have that $\check{H}^ p(\mathcal{U}, -) = R^ p\check{H}^0(\mathcal{U}, -)$ as functors on $\textit{PAb}(\mathcal{C})$ by Lemma 21.9.6. Putting everything together gives the lemma. $\square$

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Your email address will not be published.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