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

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

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

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)