The Stacks project

Lemma 24.22.6. 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 a Grothendieck abelian category.

Proof. By Lemma 24.13.2 and the definition of a Grothendieck abelian category (Injectives, Definition 19.10.1) it suffices to show that $\textit{Mod}(\mathcal{A}, \text{d})$ has a generator. For every object $U$ of $\mathcal{C}$ we denote $C_ U$ the cone on the identity map $\mathcal{A}_ U \to \mathcal{A}_ U$ as in Remark 24.22.5. We claim that

\[ \mathcal{G} = \bigoplus \nolimits _{k, U} j_{U!}C_ U[k] \]

is a generator where the sum is over all objects $U$ of $\mathcal{C}$ and $k \in \mathbf{Z}$. Indeed, given a differential graded $\mathcal{A}$-module $\mathcal{M}$ if there are no nonzero maps from $\mathcal{G}$ to $\mathcal{M}$, then we see that for all $k$ and $U$ we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A})}(j_{U!}C_ U[k], \mathcal{M}) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A}_ U)}(C_ U[k], \mathcal{M}|_ U) \\ & = \{ (x, y) \in \mathcal{M}^{-k}(U) \times \mathcal{M}^{-k - 1}(U) \mid x = \text{d}(y)\} \end{align*}

is equal to zero. Hence $\mathcal{M}$ is zero. $\square$

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