Lemma 20.11.5. Let X be a ringed space. Let \mathcal{U} : U = \bigcup _{i \in I} U_ i be an open covering. For any sheaf of \mathcal{O}_ X-modules \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{Mod}(\mathcal{O}_ X) \to \textit{PMod}(\mathcal{O}_ X) \quad \text{and}\quad \check{H}^0(\mathcal{U}, - ) : \textit{PMod}(\mathcal{O}_ X) \to \text{Mod}_{\mathcal{O}_ X(U)}.
Namely, we have \check{H}^0(\mathcal{U}, i(\mathcal{F})) = \mathcal{F}(U) by Lemma 20.9.2. We have that i(\mathcal{I}) is Čech acyclic by Lemma 20.11.1. And we have that \check{H}^ p(\mathcal{U}, -) = R^ p\check{H}^0(\mathcal{U}, -) as functors on \textit{PMod}(\mathcal{O}_ X) by Lemma 20.10.5. Putting everything together gives the lemma.
\square
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