The Stacks project

Lemma 74.16.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \subset |X|$ be a closed subset such that $|X| \setminus T$ is quasi-compact. An object of $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ is compact if and only if it is perfect as an object of $D(\mathcal{O}_ X)$.

Proof. We observe that $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ is a triangulated category with direct sums by the remark preceding the lemma. By Proposition 74.16.1 the perfect objects define compact objects of $D(\mathcal{O}_ X)$ hence a fortiori of any subcategory preserved under taking direct sums. For the converse we will use there exists a generator $E \in D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ which is a perfect complex of $\mathcal{O}_ X$-modules, see Lemma 74.15.6. Hence by the above, $E$ is compact. Then it follows from Derived Categories, Proposition 13.37.6 that $E$ is a classical generator of the full subcategory of compact objects of $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$. Thus any compact object can be constructed out of $E$ by a finite sequence of operations consisting of (a) taking shifts, (b) taking finite direct sums, (c) taking cones, and (d) taking direct summands. Each of these operations preserves the property of being perfect and the result follows. $\square$


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