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The Stacks project

Lemma 21.44.9. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{E}^\bullet , \mathcal{F}^\bullet be complexes of \mathcal{O}-modules with \mathcal{E}^\bullet strictly perfect. Then the internal hom R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}^\bullet , \mathcal{F}^\bullet ) is represented by the complex \mathcal{H}^\bullet with terms

\mathcal{H}^ n = \bigoplus \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{E}^{-q}, \mathcal{F}^ p)

and differential as described in Section 21.35.

Proof. Choose a quasi-isomorphism \mathcal{F}^\bullet \to \mathcal{I}^\bullet into a K-injective complex. Let (\mathcal{H}')^\bullet be the complex with terms

(\mathcal{H}')^ n = \prod \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}^{-q}, \mathcal{I}^ p)

which represents R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}^\bullet , \mathcal{F}^\bullet ) by the construction in Section 21.35. It suffices to show that the map

\mathcal{H}^\bullet \longrightarrow (\mathcal{H}')^\bullet

is a quasi-isomorphism. Given an object U of \mathcal{C} we have by inspection

H^0(\mathcal{H}^\bullet (U)) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ U)}(\mathcal{E}^\bullet |_ U, \mathcal{I}^\bullet |_ U) \to H^0((\mathcal{H}')^\bullet (U)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{E}^\bullet |_ U, \mathcal{I}^\bullet |_ U)

By Lemma 21.44.8 the sheafification of U \mapsto H^0(\mathcal{H}^\bullet (U)) is equal to the sheafification of U \mapsto H^0((\mathcal{H}')^\bullet (U)). A similar argument can be given for the other cohomology sheaves. Thus \mathcal{H}^\bullet is quasi-isomorphic to (\mathcal{H}')^\bullet which proves the lemma. \square


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