Lemma 34.8.12. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. Let $\mathcal{P}$ be one of the properties of modules^{1} 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 34.8.11 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 34.7.1, 34.7.3, 34.7.4, 34.7.5, and 34.7.6. 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 29.9.1) hence this reduces to the case of finite type modules (details omitted).
$\square$

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