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The Stacks project

Remark 20.25.3. Let (X, \mathcal{O}_ X) be a ringed space. Let \mathcal{U} : X = \bigcup _{i \in I} U_ i be an open covering. Let \mathcal{F}^\bullet be a bounded below complex of \mathcal{O}_ X-modules. Let b be an integer. We claim there is a commutative diagram

\xymatrix{ \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet ))[b] \ar[r] \ar[d]_\gamma & R\Gamma (X, \mathcal{F}^\bullet )[b] \ar[d] \\ \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet [b])) \ar[r] & R\Gamma (X, \mathcal{F}^\bullet [b]) }

in the derived category where the map \gamma is the map on complexes constructed in Homology, Remark 12.18.5. This makes sense because the double complex \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet [b]) is clearly the same as the double complex \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet )[0, b] introduced in Homology, Remark 12.18.5. To check that the diagram commutes, we may choose an injective resolution \mathcal{F}^\bullet \to \mathcal{I}^\bullet as in the proof of Lemma 20.25.1. Chasing diagrams, we see that it suffices to check the diagram commutes when we replace \mathcal{F}^\bullet by \mathcal{I}^\bullet . Then we consider the extended diagram

\xymatrix{ \Gamma (X, \mathcal{I}^\bullet )[b] \ar[r] \ar[d] & \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet ))[b] \ar[r] \ar[d]_\gamma & R\Gamma (X, \mathcal{I}^\bullet )[b] \ar[d] \\ \Gamma (X, \mathcal{I}^\bullet [b]) \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet [b])) \ar[r] & R\Gamma (X, \mathcal{I}^\bullet [b]) }

where the left horizontal arrows are (20.25.0.1). Since in this case the horizontal arrows are isomorphisms in the derived category (see proof of Lemma 20.25.1) it suffices to show that the left square commutes. This is true because the map \gamma uses the sign 1 on the summands \check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^{q + b}), see formula in Homology, Remark 12.18.5.


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