The Stacks project

Lemma 36.10.1. Let $X$ be a scheme. If $E$ is an $m$-pseudo-coherent object of $D(\mathcal{O}_ X)$, then $H^ i(E)$ is a quasi-coherent $\mathcal{O}_ X$-module for $i > m$ and $H^ m(E)$ is a quotient of a quasi-coherent $\mathcal{O}_ X$-module. If $E$ is pseudo-coherent, then $E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Proof. Locally on $X$ there exists a strictly perfect complex $\mathcal{E}^\bullet $ such that $H^ i(E)$ is isomorphic to $H^ i(\mathcal{E}^\bullet )$ for $i > m$ and $H^ m(E)$ is a quotient of $H^ m(\mathcal{E}^\bullet )$. The sheaves $\mathcal{E}^ i$ are direct summands of finite free modules, hence quasi-coherent. The lemma follows. $\square$

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