Lemma 20.33.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Set $A = \Gamma (X, \mathcal{O}_ X)$. Suppose that $X = U \cup V$ is a union of two open subsets. For objects $K$ and $M$ of $D(\mathcal{O}_ X)$ we have a map of distinguished triangles

\[ \xymatrix{ R\Gamma (X, K) \otimes _ A^\mathbf {L} R\Gamma (X, M) \ar[r] \ar[d] & R\Gamma (X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \ar[d] \\ R\Gamma (X, K) \otimes _ A^\mathbf {L} (R\Gamma (U, M) \oplus R\Gamma (V, M)) \ar[r] \ar[d] & R\Gamma (U, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \oplus R\Gamma (V, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M)) \ar[d] \\ R\Gamma (X, K) \otimes _ A^\mathbf {L} R\Gamma (U \cap V, M) \ar[r] \ar[d] & R\Gamma (U \cap V, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \ar[d] \\ R\Gamma (X, K) \otimes _ A^\mathbf {L} R\Gamma (X, M)[1] \ar[r] & R\Gamma (X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M)[1] } \]

where

the horizontal arrows are given by cup product,

on the right hand side we have the distinguished triangle of Lemma 20.33.4 for $K \otimes _{\mathcal{O}_ X}^\mathbf {L} M$, and

on the left hand side we have the exact functor $R\Gamma (X, K) \otimes _ A^\mathbf {L} - $ applied to the distinguished triangle of Lemma 20.33.4 for $M$.

**Proof.**
Choose a K-flat complex $T^\bullet $ of flat $A$-modules representing $R\Gamma (X, K)$, see More on Algebra, Lemma 15.59.10. Denote $T^\bullet \otimes _ A \mathcal{O}_ X$ the pullback of $T^\bullet $ by the morphism of ringed spaces $(X, \mathcal{O}_ X) \to (pt, A)$. There is a natural adjunction map $\epsilon : T^\bullet \otimes _ A \mathcal{O}_ X \to K$ in $D(\mathcal{O}_ X)$. Observe that $T^\bullet \otimes _ A \mathcal{O}_ X$ is a K-flat complex of $\mathcal{O}_ X$-modules with flat terms, see Lemma 20.26.8 and Modules, Lemma 17.20.2. By Lemma 20.26.17 we can find a morphism of complexes

\[ T^\bullet \otimes _ A \mathcal{O}_ X \longrightarrow \mathcal{K}^\bullet \]

of $\mathcal{O}_ X$-modules representing $\epsilon $ such that $\mathcal{K}^\bullet $ is a K-flat complex with flat terms. Namely, by the construction of $D(\mathcal{O}_ X)$ we can first represent $\epsilon $ by some map of complexes $e : T^\bullet \otimes _ A \mathcal{O}_ X \to \mathcal{L}^\bullet $ of $\mathcal{O}_ X$-modules representing $\epsilon $ and then we can apply the lemma to $e$. Choose a K-injective complex $\mathcal{I}^\bullet $ whose terms are injective $\mathcal{O}_ X$-modules representing $M$. Finally, choose a quasi-isomorphism

\[ \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{I}^\bullet ) \longrightarrow \mathcal{J}^\bullet \]

into a K-injective complex whose terms are injective $\mathcal{O}_ X$-modules. Observe that source and target of this arrow represent $K \otimes _{\mathcal{O}_ X}^\mathbf {L} M$ in $D(\mathcal{O}_ X)$. At this point, for any open $W \subset X$ we obtain a map of complexes

\[ \text{Tot}(T^\bullet \otimes _ A \mathcal{I}^\bullet (W)) \to \text{Tot}(\mathcal{K}^\bullet (W) \otimes _ A \mathcal{I}^\bullet (W)) \to \mathcal{J}^\bullet (W) \]

of $A$-modules whose composition represents the map

\[ R\Gamma (X, K) \otimes _ A^\mathbf {L} R\Gamma (W, M) \longrightarrow R\Gamma (W, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \]

in $D(A)$. Clearly, these maps are compatible with restriction mappings. OK, so now we can consider the following commutative(!) diagram of complexes of $A$-modules

\[ \xymatrix{ 0 \ar[d] & 0 \ar[d] \\ \text{Tot}(T^\bullet \otimes _ A \mathcal{I}^\bullet (X)) \ar[d] \ar[r] & \mathcal{J}^\bullet (X) \ar[d] \\ \text{Tot}(T^\bullet \otimes _ A (\mathcal{I}^\bullet (U) \oplus \mathcal{I}^\bullet (V)) \ar[d] \ar[r] & \mathcal{J}^\bullet (U) \oplus \mathcal{J}^\bullet (V) \ar[d] \\ \text{Tot}(T^\bullet \otimes _ A \mathcal{I}^\bullet (U \cap V)) \ar[r] \ar[d] & \mathcal{J}^\bullet (U \cap V) \ar[d] \\ 0 & 0 } \]

By the proof of Lemma 20.8.2 the columns are exact sequences of complexes of $A$-modules (this also uses that $\text{Tot}(T^\bullet \otimes _ A -)$ transforms short exact sequences of complexes of $A$-modules into short exact sequences as the terms of $T^\bullet $ are flat $A$-modules). Since the distinguished triangles of Lemma 20.33.4 are the distinguished triangles associated to these short exact sequences of complexes, the desired result follows from the functoriality of “taking the associated distinguished triangle” discussed in Derived Categories, Section 13.12.
$\square$

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