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 horizonal 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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