The Stacks project

21.36 Global derived hom

Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $K, L \in D(\mathcal{O})$. Using the construction of the internal hom in the derived category we obtain a well defined object

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

in $D(\Gamma (\mathcal{C}, \mathcal{O}))$. By Lemma 21.35.1 we have

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

and

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

If $f : (\mathcal{C}', \mathcal{O}') \to (\mathcal{C}, \mathcal{O})$ is a morphism of ringed topoi, then there is a canonical map

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

in $D(\Gamma (\mathcal{O}))$ by taking global sections of the map defined in Remark 21.35.11.

Comments (2)

Comment #7487 by Hao Peng on

The appearence of X is weird. Should be

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