Lemma 35.10.4. Let S be a scheme. Let \tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} . The category \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}) of quasi-coherent modules on (\mathit{Sch}/S)_\tau has the following properties:
Any direct sum of quasi-coherent sheaves is quasi-coherent.
Any colimit of quasi-coherent sheaves is quasi-coherent.
The cokernel of a morphism of quasi-coherent sheaves is quasi-coherent.
Given a short exact sequence of \mathcal{O}-modules 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 if \mathcal{F}_1 and \mathcal{F}_3 are quasi-coherent so is \mathcal{F}_2.
Given two quasi-coherent \mathcal{O}-modules the tensor product is quasi-coherent.
Given two quasi-coherent \mathcal{O}-modules \mathcal{F}, \mathcal{G} such that \mathcal{F} is finite locally free, the internal hom \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) is quasi-coherent.
Proof.
The corresponding facts hold for quasi-coherent modules on the scheme S, see Schemes, Section 26.24. The proof will be to use Lemma 35.10.2 to transfer these truths to (\mathit{Sch}/S)_\tau .
Proof of (1). Let \mathcal{F}_ i, i \in I be a family of objects of \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}). Write \mathcal{F}_ i = \mathcal{G}_ i^ a for some quasi-coherent modules \mathcal{G}_ i on S. Then \bigoplus \mathcal{F}_ i = (\bigoplus \mathcal{G}_ i)^ a by the lemma cited and we conclude.
Proof of (2). Let \mathcal{I} \to \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}), i \mapsto \mathcal{F}_ i be a diagram. Write \mathcal{F}_ i = \mathcal{G}_ i^ a so we get a diagram \mathcal{I} \to \mathit{QCoh}(\mathcal{O}_ S). Then \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i = (\mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i)^ a by the lemma cited and we conclude.
Proof of (3). Let a : \mathcal{F} \to \mathcal{F}' be an arrow of \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}). Write a = b^ a for some map b : \mathcal{G} \to \mathcal{G}' of quasi-coherent modules on S. By the lemma cited we have \mathop{\mathrm{Coker}}(a) = \mathop{\mathrm{Coker}}(b)^ a (because a cokernel is a colimit) and we conclude.
Proof of (4). Write \mathcal{F}_1 = \mathcal{G}_1^ a and \mathcal{F}_3 = \mathcal{G}_3^ a with \mathcal{G}_ i quasi-coherent on S. By Lemma 35.10.2 part (10) we conclude.
Proof of (5). Let \mathcal{F} and \mathcal{F}' be in \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}). Write \mathcal{F} = \mathcal{G}^ a and \mathcal{F}' = (\mathcal{G}')^ a with \mathcal{G} and \mathcal{G}' quasi-coherent on S. By the lemma cited we have \mathcal{F} \otimes _\mathcal {O} \mathcal{F}' = (\mathcal{G} \otimes _{\mathcal{O}_ S} \mathcal{G}')^ a and we conclude.
Proof of (6). Write \mathcal{F} = \mathcal{H}^ a for some quasi-coherent \mathcal{O}_ S-module. By Lemma 35.8.10 we see that \mathcal{H} is finite locally free. The problem is Zariski local on S (details omitted) hence we may assume \mathcal{H} = \mathcal{O}_ S^{\oplus n} is finite free. Then \mathcal{F} = \mathcal{O}^{\oplus n} and \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) = \mathcal{G}^{\oplus n} is quasi-coherent.
\square
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