Lemma 24.26.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The full subcategory $\text{Ac}$ of the homotopy category $K(\textit{Mod}(\mathcal{A}, \text{d}))$ consisting of acyclic modules is a strictly full saturated triangulated subcategory of $K(\textit{Mod}(\mathcal{A}, \text{d}))$.

**Proof.**
Of course an object $\mathcal{M}$ of $K(\textit{Mod}(\mathcal{A}, \text{d}))$ is in $\text{Ac}$ if and only if $H^ i(\mathcal{M}) = H^0(\mathcal{M}[i])$ is zero for all $i$. The lemma follows from this, Lemma 24.26.1, and Derived Categories, Lemma 13.6.3. See also Derived Categories, Definitions 13.6.1 and 13.3.4 and Lemma 13.4.16.
$\square$

## 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Your email address will not be published.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)