Remark 20.31.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, M$ be objects of $D(\mathcal{O}_ X)$. Set $A = \Gamma (X, \mathcal{O}_ X)$. Given $\xi \in H^ i(X, K)$ we get an associated map

$\xi = \xi \cup -'' : R\Gamma (X, M)[-i] \to R\Gamma (X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M)$

by representing $\xi$ as a map $\xi : A[-i] \to R\Gamma (X, K)$ as in the proof of Lemma 20.31.1 and then using the composition

$R\Gamma (X, M)[-i] = A[-i] \otimes _ A^\mathbf {L} R\Gamma (X, M) \xrightarrow {\xi \otimes 1} R\Gamma (X, K) \otimes _ A^\mathbf {L} R\Gamma (X, M) \to R\Gamma (X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M)$

where the second arrow is the global cup product $\mu$ above. On cohomology this recovers the cup product by $\xi$ as is clear from Lemma 20.31.1 and its proof.

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