The Stacks project

Lemma 21.11.6. 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 21.9.2. We have that $i(\mathcal{I})$ is Čech acyclic by Lemma 21.11.2. 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 21.10.6. Putting everything together gives the lemma. $\square$


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