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:

1. Any direct sum of quasi-coherent sheaves is quasi-coherent.

2. Any colimit of quasi-coherent sheaves is quasi-coherent.

3. The cokernel of a morphism of quasi-coherent sheaves is quasi-coherent.

4. 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 $\mathcal{F}_2$.

5. Given two quasi-coherent $\mathcal{O}$-modules the tensor product is quasi-coherent.

6. 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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