The Stacks project

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

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

defined by

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. $\square$

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

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

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

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

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. $\square$