Lemma 20.26.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be an open covering. For a bounded below complex $\mathcal{F}^\bullet $ of $\mathcal{O}_ X$-modules there is a canonical map

\[ \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )) \longrightarrow R\Gamma (X, \mathcal{F}^\bullet ) \]

functorial in $\mathcal{F}^\bullet $ and compatible with (20.26.0.1) and (20.26.0.2). There is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with

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

converging to $H^{p + q}(X, \mathcal{F}^\bullet )$.

**Proof.**
Let ${\mathcal I}^\bullet $ be a bounded below complex of injectives. The map (20.26.0.1) for $\mathcal{I}^\bullet $ is a map $\Gamma (X, {\mathcal I}^\bullet ) \to \text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^\bullet ))$. This is a quasi-isomorphism of complexes of abelian groups as follows from Homology, Lemma 12.22.7 applied to the double complex $\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^\bullet )$ using Lemma 20.12.1. Suppose ${\mathcal F}^\bullet \to {\mathcal I}^\bullet $ is a quasi-isomorphism of ${\mathcal F}^\bullet $ into a bounded below complex of injectives. Since $R\Gamma (X, {\mathcal F}^\bullet )$ is represented by the complex $\Gamma (X, {\mathcal I}^\bullet )$ we obtain the map of the lemma using

\[ \text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal F}^\bullet )) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^\bullet )). \]

We omit the verification of functoriality and compatibilities. To construct the spectral sequence of the lemma, choose a Cartan-Eilenberg resolution $\mathcal{F}^\bullet \to \mathcal{I}^{\bullet , \bullet }$, see Derived Categories, Lemma 13.21.2. In this case $\mathcal{F}^\bullet \to \text{Tot}(\mathcal{I}^{\bullet , \bullet })$ is an injective resolution and hence

\[ \text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, \text{Tot}({\mathcal I}^{\bullet , \bullet }))) \]

computes $R\Gamma (X, \mathcal{F}^\bullet )$ as we've seen above. By Homology, Remark 12.22.8 we can view this as the total complex associated to the triple complex $\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^{\bullet , \bullet })$ hence, using the same remark we can view it as the total complex associate to the double complex $A^{\bullet , \bullet }$ with terms

\[ A^{n, m} = \bigoplus \nolimits _{p + q = n} \check{\mathcal{C}}^ p({\mathcal U}, \mathcal{I}^{q, m}) \]

Since $\mathcal{I}^{q, \bullet }$ is an injective resolution of $\mathcal{F}^ q$ we can apply the first spectral sequence associated to $A^{\bullet , \bullet }$ (Homology, Lemma 12.22.4) to get a spectral sequence with

\[ E_1^{n, m} = \bigoplus \nolimits _{p + q = n} \check{\mathcal{C}}^ p(\mathcal{U}, \underline{H}^ m(\mathcal{F}^ q)) \]

which is the $n$th term of the complex $\text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \underline{H}^ m(\mathcal{F}^\bullet ))$. Hence we obtain $E_2$ terms as described in the lemma. Convergence by Homology, Lemma 12.22.6.
$\square$

## Comments (2)

Comment #3217 by Ingo Blechschmidt on

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