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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