The Stacks project

Lemma 20.31.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{K}^\bullet $ and $\mathcal{M}^\bullet $ be bounded below complexes of $\mathcal{O}_ X$-modules. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be an open covering Then

\[ \xymatrix{ \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{K}^\bullet )) \otimes _ A^\mathbf {L} \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{M}^\bullet )) \ar[d] \ar[r] & R\Gamma (X, \mathcal{K}^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (X, \mathcal{M}^\bullet ) \ar[d]^\mu \\ \text{Tot}( \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{K}^\bullet )) \otimes _ A \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{M}^\bullet ))) \ar[d]^{(07MB)} & R\Gamma (X, \mathcal{K}^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{M}^\bullet ) \ar[d] \\ \text{Tot}( \check{\mathcal{C}}^\bullet ({\mathcal U}, \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{M}^\bullet ) )) \ar[r] & R\Gamma (X, \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{M}^\bullet )) } \]

where the horizontal arrows are the ones in Lemma 20.25.1 commutes in $D(A)$.

Proof. Choose quasi-isomorphisms of complexes $a : \mathcal{K}^\bullet \to \mathcal{K}_1^\bullet $ and $b : \mathcal{M}^\bullet \to \mathcal{M}_1^\bullet $ as in Lemma 20.30.2. Since the maps $a$ and $b$ on stalks are homotopy equivalences we see that the induced map

\[ \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{M}^\bullet ) \to \text{Tot}(\mathcal{K}_1^\bullet \otimes _{\mathcal{O}_ X} \mathcal{M}_1^\bullet ) \]

is a homotopy equivalence on stalks too (More on Algebra, Lemma 15.57.1) and hence a quasi-isomorphism. Thus the targets

\[ R\Gamma (X, \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{M}^\bullet )) = R\Gamma (X, \text{Tot}(\mathcal{K}_1^\bullet \otimes _{\mathcal{O}_ X} \mathcal{M}_1^\bullet )) \]

of the two diagrams are the same in $D(A)$. It follows that it suffices to prove the diagram commutes for $\mathcal{K}$ and $\mathcal{M}$ replaced by $\mathcal{K}_1$ and $\mathcal{M}_1$. This reduces us to the case discussed in the next paragraph.

Assume $\mathcal{K}^\bullet $ and $\mathcal{M}^\bullet $ are bounded below complexes of flasque $\mathcal{O}_ X$-modules and consider the diagram relating the cup product with the cup product ( on Čech complexes. Then we can consider the commutative diagram

\[ \xymatrix{ \Gamma (X, \mathcal{K}^\bullet ) \otimes _ A^\mathbf {L} \Gamma (X, \mathcal{M}^\bullet ) \ar[d] \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{K}^\bullet )) \otimes _ A^\mathbf {L} \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{M}^\bullet )) \ar[d] \\ \text{Tot}(\Gamma (X, \mathcal{K}^\bullet ) \otimes _ A \Gamma (X, \mathcal{M}^\bullet )) \ar[d] \ar[r] & \text{Tot}( \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{K}^\bullet )) \otimes _ A \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{M}^\bullet ))) \ar[d]^{(07MB)} \\ \Gamma (X, \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{M}^\bullet )) \ar[r] & \text{Tot}( \check{\mathcal{C}}^\bullet ({\mathcal U}, \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{M}^\bullet ) )) } \]

In this diagram the horizontal arrows are isomorphisms in $D(A)$ because for a bounded below complex of flasque modules such as $\mathcal{K}^\bullet $ we have

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

in $D(A)$. This follows from Lemma 20.12.3, Derived Categories, Lemma 13.16.7, and Lemma 20.25.2. Hence the commutativity of the diagram of the lemma involving ( follows from the already proven commutativity of Lemma 20.31.2 where $f$ is the morphism to a point (see discussion following Lemma 20.31.2). $\square$

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