Lemma 35.8.7. Let S be a scheme. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ S-module. Let \tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} .
The sheaf \mathcal{F}^ a is a quasi-coherent \mathcal{O}-module on (\mathit{Sch}/S)_\tau , as defined in Modules on Sites, Definition 18.23.1.
If \tau = Zariski or \tau = {\acute{e}tale}, then the sheaf \mathcal{F}^ a is a quasi-coherent \mathcal{O}-module on S_{Zar} or S_{\acute{e}tale} as defined in Modules on Sites, Definition 18.23.1.
Proof.
Let \{ S_ i \to S\} be a Zariski covering such that we have exact sequences
\bigoplus \nolimits _{k \in K_ i} \mathcal{O}_{S_ i} \longrightarrow \bigoplus \nolimits _{j \in J_ i} \mathcal{O}_{S_ i} \longrightarrow \mathcal{F} \longrightarrow 0
for some index sets K_ i and J_ i. This is possible by the definition of a quasi-coherent sheaf on a ringed space (See Modules, Definition 17.10.1).
Proof of (1). Let \tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} . It is clear that \mathcal{F}^ a|_{(\mathit{Sch}/S_ i)_\tau } also sits in an exact sequence
\bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \mathcal{F}^ a|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow 0
Hence \mathcal{F}^ a is quasi-coherent by Modules on Sites, Lemma 18.23.3.
Proof of (2). Let \tau = {\acute{e}tale}. It is clear that \mathcal{F}^ a|_{(S_ i)_{\acute{e}tale}} also sits in an exact sequence
\bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(S_ i)_{\acute{e}tale}} \longrightarrow \bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(S_ i)_{\acute{e}tale}} \longrightarrow \mathcal{F}^ a|_{(S_ i)_{\acute{e}tale}} \longrightarrow 0
Hence \mathcal{F}^ a is quasi-coherent by Modules on Sites, Lemma 18.23.3. The case \tau = Zariski is similar (actually, it is really tautological since the corresponding ringed topoi agree).
\square
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