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 $G : \textit{Mod}(\mathcal{A}) \to \textit{Mod}(\mathcal{A}, \text{d})$.
24.24 The differential graded hull of a graded module
The differential graded hull of a graded module $\mathcal{N}$ is the result of applying the functor $G$ in the following lemma.
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. $\square$
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module and suppose we have a short exact sequence
\[ 0 \to \mathcal{N} \to F(\mathcal{M}) \to \mathcal{N}' \to 0 \]
in $\textit{Mod}(\mathcal{A})$. Then we obtain a canonical graded $\mathcal{A}$-module homomorphism
\[ \overline{\text{d}} : \mathcal{N} \to \mathcal{N}'[1] \]
as follows: given a local section $x$ of $\mathcal{N}$ denote $\overline{\text{d}}(x)$ the image in $\mathcal{N}'$ of $\text{d}_\mathcal {M}(x)$ when $x$ is viewed as a local section of $\mathcal{M}$.
Lemma 24.24.2. The functors $F, G$ of Lemma 24.24.1 have the following properties. Given a graded $\mathcal{A}$-module $\mathcal{N}$ we have
the counit $\mathcal{N} \to F(G(\mathcal{N}))$ is injective,
the map $\overline{\text{d}} : \mathcal{N} \to \mathop{\mathrm{Coker}}(\mathcal{N} \to F(G(\mathcal{N})))[1]$ is an isomorphism, and
$G(\mathcal{N})$ is an acyclic differential graded $\mathcal{A}$-module.
Proof. We observe that property (3) is a consequence of properties (1) and (2). Namely, if $s$ is a nonzero local section of $F(G(\mathcal{N}))$ with $\text{d}(s) = 0$, then $s$ cannot be in the image of $\mathcal{N} \to F(G(\mathcal{N}))$. Hence we can write the image $\overline{s}$ of $s$ in the cokernel as $\overline{\text{d}}(s')$ for some local section $s'$ of $\mathcal{N}$. Then we see that $s = \text{d}(s')$ because the difference $s - \text{d}(s')$ is still in the kernel of $\text{d}$ and is contained in the image of the counit.
Let us write temporarily $\mathcal{A}_{gr}$, respectively $\mathcal{A}_{dg}$ the sheaf $\mathcal{A}$ viewed as a (right) graded module over itself, respectively as a (right) differential graded module over itself. The most important case of the lemma is to understand what is $G(\mathcal{A}_{gr})$. Of course $G(\mathcal{A}_{gr})$ is the object of $\textit{Mod}(\mathcal{A}, \text{d})$ representing the functor
\[ \mathcal{M} \longmapsto \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A})}(\mathcal{A}_{gr}, F(\mathcal{M})) = \Gamma (\mathcal{C}, \mathcal{M}) \]
By Remark 24.22.5 we see that this functor represented by $C[-1]$ where $C$ is the cone on the identity of $\mathcal{A}_{dg}$. We have a short exact sequence
\[ 0 \to \mathcal{A}_{dg}[-1] \to C[-1] \to \mathcal{A}_{dg} \to 0 \]
in $\textit{Mod}(\mathcal{A}, \text{d})$ which is split by the counit $\mathcal{A}_{gr} \to F(C[-1])$ in $\textit{Mod}(\mathcal{A})$. Thus $G(\mathcal{A}_{gr})$ satisfies properties (1) and (2).
Let $U$ be an object of $\mathcal{C}$. Denote $j_ U : \mathcal{C}/U \to \mathcal{C}$ the localization morphism. Denote $\mathcal{A}_ U$ the restriction of $\mathcal{A}$ to $U$. We will use the notation $\mathcal{A}_{U, gr}$ to denote $\mathcal{A}_ U$ viewed as a graded $\mathcal{A}_ U$-module. Denote $F_ U : \textit{Mod}(\mathcal{A}_ U, \text{d}) \to \textit{Mod}(\mathcal{A}_ U)$ the forgetful functor and denote $G_ U$ its adjoint. Then we have the commutative diagrams
\[ \vcenter { \xymatrix{ \textit{Mod}(\mathcal{A}, \text{d}) \ar[d]_{j_ U^*} \ar[r]_ F & \textit{Mod}(\mathcal{A}) \ar[d]^{j_ U^*} \\ \textit{Mod}(\mathcal{A}_ U, \text{d}) \ar[r]^{F_ U} & \textit{Mod}(\mathcal{A}_ U) } } \quad \text{and}\quad \vcenter { \xymatrix{ \textit{Mod}(\mathcal{A}_ U, \text{d}) \ar[r]_{F_ U} \ar[d]_{j_{U!}} & \textit{Mod}(\mathcal{A}_ U) \ar[d]^{j_{U!}} \\ \textit{Mod}(\mathcal{A}, \text{d}) \ar[r]^ F & \textit{Mod}(\mathcal{A}) } } \]
by the construction of $j^*_ U$ and $j_{U!}$ in Sections 24.9, 24.18, 24.10, and 24.19. By uniqueness of adjoints we obtain $j_{U!} \circ G_ U = G \circ j_{U!}$. Since $j_{U!}$ is an exact functor, we see that the properties (1) and (2) for the counit $\mathcal{A}_{U, gr} \to F_ U(G_ U(\mathcal{A}_{U, gr}))$ which we've seen in the previous part of the proof imply properties (1) and (2) for the counit $j_{U!}\mathcal{A}_{U, gr} \to F(G(j_{U!}\mathcal{A}_{U, gr})) = j_{U!}F_ U(G_ U(\mathcal{A}_{U, gr}))$.
In the proof of Lemma 24.11.1 we have seen that any object of $\textit{Mod}(\mathcal{A})$ is a quotient of a direct sum of copies of $j_{U!}\mathcal{A}_{U, gr}$. Since $G$ is a left adjoint, we see that $G$ commutes with direct sums. Thus properties (1) and (2) hold for direct sums of objects for which they hold. Thus we see that every object $\mathcal{N}$ of $\textit{Mod}(\mathcal{A})$ fits into an exact sequence
\[ \mathcal{N}_1 \to \mathcal{N}_0 \to \mathcal{N} \to 0 \]
such that (1) and (2) hold for $\mathcal{N}_1$ and $\mathcal{N}_0$. We leave it to the reader to deduce (1) and (2) for $\mathcal{N}$ using that $G$ is right exact. $\square$
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