Lemma 20.41.7. Let (X, \mathcal{O}_ X) be a ringed space. Let (\mathcal{I}')^\bullet \to \mathcal{I}^\bullet be a quasi-isomorphism of K-injective complexes of \mathcal{O}_ X-modules. Let (\mathcal{L}')^\bullet \to \mathcal{L}^\bullet be a quasi-isomorphism of complexes of \mathcal{O}_ X-modules. Then
\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , (\mathcal{I}')^\bullet ) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet ((\mathcal{L}')^\bullet , \mathcal{I}^\bullet )
is a quasi-isomorphism.
Proof.
Let M be the object of D(\mathcal{O}_ X) represented by \mathcal{I}^\bullet and (\mathcal{I}')^\bullet . Let L be the object of D(\mathcal{O}_ X) represented by \mathcal{L}^\bullet and (\mathcal{L}')^\bullet . By Lemma 20.41.6 we see that the sheaves
H^0(\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , (\mathcal{I}')^\bullet )) \quad \text{and}\quad H^0(\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet ((\mathcal{L}')^\bullet , \mathcal{I}^\bullet ))
are both equal to the sheaf associated to the presheaf
U \longmapsto \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(L|_ U, M|_ U)
Thus the map is a quasi-isomorphism.
\square
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