The Stacks project

Lemma 21.33.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{I}')^\bullet \to \mathcal{I}^\bullet $ be a quasi-isomorphism of K-injective complexes of $\mathcal{O}$-modules. Let $(\mathcal{L}')^\bullet \to \mathcal{L}^\bullet $ be a quasi-isomorphism of complexes of $\mathcal{O}$-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 $D(\mathcal{O})$ represented by $\mathcal{I}^\bullet $ and $(\mathcal{I}')^\bullet $. Let $L$ be the object of $D(\mathcal{O})$ represented by $\mathcal{L}^\bullet $ and $(\mathcal{L}')^\bullet $. By Lemma 21.33.6 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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