Lemma 24.25.10. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let (\mathcal{A}, \text{d}) be a sheaf of differential graded algebras on (\mathcal{C}, \mathcal{O}). Let \mathcal{I} be a K-injective and graded injective object of \textit{Mod}(\mathcal{A}, \text{d}). For every solid diagram in \textit{Mod}(\mathcal{A}, \text{d})
\xymatrix{ \mathcal{M} \ar[r]_ a \ar[d]_ b & \mathcal{I} \\ \mathcal{M}' \ar@{..>}[ru] }
where b is a quasi-isomorphism a dotted arrow exists making the diagram commute up to homotopy.
Proof.
After replacing \mathcal{M}' by the direct sum of \mathcal{M}' and the cone on the identity on \mathcal{M} (which is acyclic) we may assume b is also injective. Then the cokernel \mathcal{Q} of b is acyclic. Thus we see that
\mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{Q}, \mathcal{I}) = \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{Q}, \mathcal{I})[1] = 0
as \mathcal{I} is K-injective. As \mathcal{I} is graded injective by Remark 24.25.3 we see that
\mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{M}', \mathcal{I}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{M}, \mathcal{I})
is bijective and the proof is complete.
\square
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