The Stacks project

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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