The Stacks project

Lemma 24.25.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $T$ and for each $t \in T$ an injective map $\mathcal{M}_ t \to \mathcal{M}'_ t$ of acyclic differential graded $\mathcal{A}$-modules such that for an object $\mathcal{I}$ of $\textit{Mod}(\mathcal{A}, \text{d})$ the following are equivalent

  1. $\mathcal{I}$ is graded injective, and

  2. for every solid diagram

    \[ \xymatrix{ \mathcal{M}_ t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{M}'_ t \ar@{..>}[ru] } \]

    a dotted arrow exists in $\textit{Mod}(\mathcal{A}, \text{d})$ making the diagram commute.

Proof. Let $T$ and $\mathcal{N}_ t \to \mathcal{N}'_ t$ be as in Lemma 24.25.1. Denote $F : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{A})$ the forgetful functor. Let $G$ be the left adjoint functor to $F$ as in Lemma 24.24.1. Set

\[ \mathcal{M}_ t = G(\mathcal{N}_ t) \to G(\mathcal{N}'_ t) = \mathcal{M}'_ t \]

This is an injective map of acyclic differential graded $\mathcal{A}$-modules by Lemma 24.24.2. Since $G$ is the left adjoint to $F$ we see that there exists a dotted arrow in the diagram

\[ \xymatrix{ \mathcal{M}_ t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{M}'_ t \ar@{..>}[ru] } \]

if and only if there exists a dotted arrow in the diagram

\[ \xymatrix{ \mathcal{N}_ t \ar[r] \ar[d] & F(\mathcal{I}) \\ \mathcal{N}'_ t \ar@{..>}[ru] } \]

Hence the result follows from the choice of our collection of arrows $\mathcal{N}_ t \to \mathcal{N}_ t'$. $\square$

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