Lemma 24.24.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The forgetful functor $F : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{A})$ has a left adjoint $G : \textit{Mod}(\mathcal{A}) \to \textit{Mod}(\mathcal{A}, \text{d})$.

Proof. To prove the existence of $G$ we can use the adjoint functor theorem, see Categories, Theorem 4.25.3 (observe that we have switched the roles of $F$ and $G$). The exactness conditions on $F$ are satisfied by Lemma 24.13.2. The set theoretic condition can be seen as follows: suppose given a graded $\mathcal{A}$-module $\mathcal{N}$. Then for any map

$\varphi : \mathcal{N} \longrightarrow F(\mathcal{M})$

we can consider the smallest differential graded $\mathcal{A}$-submodule $\mathcal{M}' \subset \mathcal{M}$ with $\mathop{\mathrm{Im}}(\varphi ) \subset F(\mathcal{M}')$. It is clear that $\mathcal{M}'$ is the image of the map of graded $\mathcal{A}$-modules

$\mathcal{N} \oplus \mathcal{N}[-1] \otimes _\mathcal {O} \mathcal{A} \longrightarrow \mathcal{M}$

defined by

$(n, \sum n_ i \otimes a_ i) \longmapsto \varphi (n) + \sum \text{d}(\varphi (n_ i)) a_ i$

because the image of this map is easily seen to be a differential graded submodule of $\mathcal{M}$. Thus the number of possible isomorphism classes of these $\mathcal{M}'$ is bounded and we conclude. $\square$

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