The Stacks project

20.44 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

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(K, L) = R\Gamma (X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)) \]

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 20.42.1 we have

\[ H^0(R\mathop{\mathrm{Hom}}\nolimits _ X(K, L)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L), \quad H^ p(R\mathop{\mathrm{Hom}}\nolimits _ X(K, L)) = \mathop{\mathrm{Ext}}\nolimits _{D(\mathcal{O}_ X)}^ p(K, L) \]

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

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(K, L) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ Y(Lf^*K, Lf^*L) \]

in $D(\Gamma (X, \mathcal{O}_ X))$ by taking global sections of the map defined in Remark 20.42.13.

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