Lemma 13.37.2. Let $\mathcal{D}$ be a (pre-)triangulated category with direct sums. Then the compact objects of $\mathcal{D}$ form the objects of a Karoubian, saturated, strictly full, (pre-)triangulated subcategory $\mathcal{D}_ c$ of $\mathcal{D}$.

**Proof.**
Let $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\mathcal{D}$ with $X$ and $Y$ compact. Then it follows from Lemma 13.4.2 and the five lemma (Homology, Lemma 12.5.20) that $Z$ is a compact object too. It is clear that if $X \oplus Y$ is compact, then $X$, $Y$ are compact objects too. Hence $\mathcal{D}_ c$ is a saturated triangulated subcategory. Since $\mathcal{D}$ is Karoubian by Lemma 13.4.14 we conclude that the same is true for $\mathcal{D}_ c$.
$\square$

## 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Your email address will not be published.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)