The Stacks project

Lemma 35.8.10. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} $. Let $\mathcal{P}$ be one of the properties of modules1 defined in Modules on Sites, Definitions 18.17.1, 18.23.1, and 18.28.1. The equivalences of categories

\[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}) \quad \text{and}\quad \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}(S_\tau , \mathcal{O}) \]

defined by the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ seen in Proposition 35.8.9 have the property

\[ \mathcal{F}\text{ has }\mathcal{P} \Leftrightarrow \mathcal{F}^ a\text{ has }\mathcal{P}\text{ as an }\mathcal{O}\text{-module} \]

except (possibly) when $\mathcal{P}$ is “locally free” or “coherent”. If $\mathcal{P}=$“coherent” the equivalence holds for $\mathit{QCoh}(\mathcal{O}_ S) \to \mathit{QCoh}(S_\tau , \mathcal{O})$ when $S$ is locally Noetherian and $\tau $ is Zariski or étale.

Proof. This is immediate for the global properties, i.e., those defined in Modules on Sites, Definition 18.17.1. For the local properties we can use Modules on Sites, Lemma 18.23.3 to translate “$\mathcal{F}^ a$ has $\mathcal{P}$” into a property on the members of a covering of $X$. Hence the result follows from Lemmas 35.7.1, 35.7.3, 35.7.4, 35.7.5, and 35.7.6. Being coherent for a quasi-coherent module is the same as being of finite type over a locally Noetherian scheme (see Cohomology of Schemes, Lemma 30.9.1) hence this reduces to the case of finite type modules (details omitted). $\square$

[1] The list is: free, finite free, generated by global sections, generated by $r$ global sections, generated by finitely many global sections, having a global presentation, having a global finite presentation, locally free, finite locally free, locally generated by sections, locally generated by $r$ sections, finite type, of finite presentation, coherent, or flat.

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