The Stacks project

Lemma 36.10.2. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $M^\bullet $ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_ X)$. Then $E$ is an $m$-pseudo-coherent (resp. pseudo-coherent) as an object of $D(\mathcal{O}_ X)$ if and only if $M^\bullet $ is $m$-pseudo-coherent (resp. pseudo-coherent) as a complex of $A$-modules.

Proof. It is immediate from the definitions that if $M^\bullet $ is $m$-pseudo-coherent, so is $E$. To prove the converse, assume $E$ is $m$-pseudo-coherent. As $X = \mathop{\mathrm{Spec}}(A)$ is quasi-compact with a basis for the topology given by standard opens, we can find a standard open covering $X = D(f_1) \cup \ldots \cup D(f_ n)$ and strictly perfect complexes $\mathcal{E}_ i^\bullet $ on $D(f_ i)$ and maps $\alpha _ i : \mathcal{E}_ i^\bullet \to E|_{U_ i}$ inducing isomorphisms on $H^ j$ for $j > m$ and surjections on $H^ m$. By Cohomology, Lemma 20.43.8 after refining the open covering we may assume $\alpha _ i$ is given by a map of complexes $\mathcal{E}_ i^\bullet \to \widetilde{M^\bullet }|_{U_ i}$ for each $i$. By Modules, Lemma 17.14.6 the terms $\mathcal{E}_ i^ n$ are finite locally free modules. Hence after refining the open covering we may assume each $\mathcal{E}_ i^ n$ is a finite free $\mathcal{O}_{U_ i}$-module. From the definition it follows that $M^\bullet _{f_ i}$ is an $m$-pseudo-coherent complex of $A_{f_ i}$-modules. We conclude by applying More on Algebra, Lemma 15.63.14.

The case “pseudo-coherent” follows from the fact that $E$ is pseudo-coherent if and only if $E$ is $m$-pseudo-coherent for all $m$ (by definition) and the same is true for $M^\bullet $ by More on Algebra, Lemma 15.63.5. $\square$


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