## 20.42 Global derived hom

Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L \in D(\mathcal{O}_ X)$. Using the construction of the internal hom in the derived category we obtain a well defined object

in $D(\Gamma (X, \mathcal{O}_ X))$. We will sometimes write $R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(K, L)$ for this object. By Lemma 20.40.1 we have

If $f : Y \to X$ is a morphism of ringed spaces, then there is a canonical map

in $D(\Gamma (X, \mathcal{O}_ X))$ by taking global sections of the map defined in Remark 20.40.13.

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