Lemma 36.10.3 (08E8)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.10: Pseudo-coherent and perfect complexes
Lemma 36.10.3 (cite)

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Lemma 36.10.3. Let $X$ be a Noetherian scheme. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ . For $m \in \mathbf{Z}$ the following are equivalent

$H^ i(E)$ is coherent for $i \geq m$ and zero for $i \gg 0$ , and
$E$ is $m$ -pseudo-coherent.

In particular, $E$ is pseudo-coherent if and only if $E$ is an object of $D^-_{\textit{Coh}}(\mathcal{O}_ X)$ .

Proof. As $X$ is quasi-compact we see that in both (1) and (2) the object $E$ is bounded above. Thus the question is local on $X$ and we may assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ for some Noetherian ring $A$ . In this case $E$ corresponds to a complex of $A$ -modules $M^\bullet$ by Lemma 36.3.5. By Lemma 36.10.2 we see that $E$ is $m$ -pseudo-coherent if and only if $M^\bullet$ is $m$ -pseudo-coherent. On the other hand, $H^ i(E)$ is coherent if and only if $H^ i(M^\bullet )$ is a finite $A$ -module (Properties, Lemma 28.16.1). Thus the result follows from More on Algebra, Lemma 15.64.17. $\square$

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