The Stacks project

Lemma 13.34.2. 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 13.4.13 we conclude that the same is true for $\mathcal{D}_ c$. $\square$


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