Lemma 24.13.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a differential graded $\mathcal{O}$-algebra. The category $\textit{Mod}(\mathcal{A}, \text{d})$ is an abelian category with the following properties

1. $\textit{Mod}(\mathcal{A}, \text{d})$ has arbitrary direct sums,

2. $\textit{Mod}(\mathcal{A}, \text{d})$ has arbitrary colimits,

3. filtered colimit in $\textit{Mod}(\mathcal{A}, \text{d})$ are exact,

4. $\textit{Mod}(\mathcal{A}, \text{d})$ has arbitrary products,

5. $\textit{Mod}(\mathcal{A}, \text{d})$ has arbitrary limits.

The forgetful functor

$\textit{Mod}(\mathcal{A}, \text{d}) \longrightarrow \textit{Mod}(\mathcal{A})$

sending a differential graded $\mathcal{A}$-module to its underlying graded module commutes with all limits and colimits.

Proof. Let us denote $F : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{A})$ the functor in the statement of the lemma. Observe that the category $\textit{Mod}(\mathcal{A})$ has properties (1) – (5), see Lemma 24.4.2.

Consider a homomorphism $f : \mathcal{M} \to \mathcal{N}$ of graded $\mathcal{A}$-modules. The kernel and cokernel of $f$ as maps of graded $\mathcal{A}$-modules are additionally endowed with differentials as in Definition 24.13.1. Hence these are also the kernel and cokernel in $\textit{Mod}(\mathcal{A}, \text{d})$. Thus $\textit{Mod}(\mathcal{A}, \text{d})$ is an abelian category and taking kernels and cokernels commutes with $F$.

To prove the existence of limits and colimits it is sufficient to prove the existence of products and direct sums, see Categories, Lemmas 4.14.11 and 4.14.12. The same lemmas show that proving the commutation of limits and colimits with $F$ follows if $F$ commutes with direct sums and products.

Let $\mathcal{M}_ t$, $t \in T$ be a set of differential graded $\mathcal{A}$-modules. Then we can consider the direct sum $\bigoplus \mathcal{M}_ t$ as a graded $\mathcal{A}$-module. Since the direct sum of graded modules is done termwise, it is clear that $\bigoplus \mathcal{M}_ t$ comes endowed with a differential. The reader easily verifies that this is a direct sum in $\textit{Mod}(\mathcal{A}, \text{d})$. Similarly for products.

Observe that $F$ is an exact functor and that a complex $\mathcal{M}_1 \to \mathcal{M}_2 \to \mathcal{M}_3$ of $\textit{Mod}(\mathcal{A}, \text{d})$ is exact if and only if $F(\mathcal{M}_1) \to F(\mathcal{M}_2) \to F(\mathcal{M}_3)$ is exact in $\textit{Mod}(\mathcal{A})$. Hence we conclude that (3) holds as filtered colimits are exact in $\textit{Mod}(\mathcal{A})$. $\square$

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