Lemma 75.18.2 (0DKC)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 75: Derived Categories of Spaces
Section 75.18: Characterizing pseudo-coherent complexes, I
Lemma 75.18.2 (cite)

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Lemma 75.18.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. Let $K \in D(\mathcal{O}_ X)$ supported on $T$. The following are equivalent

$K$ is pseudo-coherent, and
$K = \text{hocolim} K_ n$ where $K_ n$ is perfect, supported on $T$, and $\tau _{\geq -n}K_ n \to \tau _{\geq -n}K$ is an isomorphism for all $n$.

Proof. The proof of this lemma is exactly the same as the proof of Lemma 75.18.1 except that in the choice of the approximations we use the triples $(T, K, m)$. $\square$

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