The Stacks project

Lemma 24.25.9. 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 injective and $\mathcal{M}$ is acyclic a dotted arrow exists making the diagram commute.

Proof. Since $\mathcal{M}$ is acyclic and $\mathcal{I}$ is K-injective, there exists a graded $\mathcal{A}$-module map $h : \mathcal{M} \to \mathcal{I}$ of degree $-1$ such that $a = \text{d}(h)$. Since $\mathcal{I}$ is graded injective and $b$ is injective, there exists a graded $\mathcal{A}$-module map $h' : \mathcal{M}' \to \mathcal{I}$ of degree $-1$ such that $h = h' \circ b$. Then we can take $a' = \text{d}(h')$ as the dotted arrow. $\square$

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