The Stacks project

Lemma 35.10.6. Let $S$ be a scheme.

  1. The category $\mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O})$ has colimits and they agree with colimits in the categories $\textit{Mod}((\mathit{Sch}/S)_{Zar}, \mathcal{O})$, $\textit{Mod}((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O})$, and $\textit{Mod}((\mathit{Sch}/S)_{fppf}, \mathcal{O})$.

  2. Given $\mathcal{F}, \mathcal{G}$ in $\mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O})$ the tensor products $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$ computed in $\textit{Mod}((\mathit{Sch}/S)_{Zar}, \mathcal{O})$, $\textit{Mod}((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O})$, or $\textit{Mod}((\mathit{Sch}/S)_{fppf}, \mathcal{O})$ agree and the common value is an object of $\mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O})$.

  3. Given $\mathcal{F}, \mathcal{G}$ in $\mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O})$ with $\mathcal{F}$ finite locally free (in fppf, or equivalently ├ętale, or equivalently Zariski topology) the internal homs $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G})$ computed in $\textit{Mod}((\mathit{Sch}/S)_{Zar}, \mathcal{O})$, $\textit{Mod}((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O})$, or $\textit{Mod}((\mathit{Sch}/S)_{fppf}, \mathcal{O})$ agree and the common value is an object of $\mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O})$.

Proof. This lemma collects the results shown above in a slightly different manner. First of all, by Lemma 35.10.4 we already know the output of the construction in (1), (2), or (3) ends up in $\mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O})$. It remains to show in each case that the result is independent of the topology used. The key to this is that the equivalence $\mathit{QCoh}(\mathcal{O}_ S) \to \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O})$, $\mathcal{F} \mapsto \mathcal{F}^ a$ of Proposition 35.8.9 is given by the same formula independent of the choice of the topology $\tau \in \{ Zariski, {\acute{e}tale}, fppf\} $.

Proof of (1). Let $\mathcal{I} \to \mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O})$, $i \mapsto \mathcal{F}_ i$ 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 $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i = (\mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i)^ a$ in $\textit{Mod}((\mathit{Sch}/S)_\tau , \mathcal{O})$ for $\tau \in \{ Zariski, {\acute{e}tale}, fppf\} $ by Lemma 35.10.2. This proves (1).

Proof of (2). Write $\mathcal{F} = \mathcal{H}^ a$ and $\mathcal{G} = (\mathcal{I})^ a$ with $\mathcal{H}$ and $\mathcal{I}$ quasi-coherent on $S$. Then $\mathcal{F} \otimes _\mathcal {O} \mathcal{G} = (\mathcal{H} \otimes _\mathcal {O} \mathcal{I})^ a$ in $\textit{Mod}((\mathit{Sch}/S)_\tau , \mathcal{O})$ for $\tau \in \{ Zariski, {\acute{e}tale}, fppf\} $ by Lemma 35.10.2. This proves (2).

Proof of (3). Let $\mathcal{F}$ and $\mathcal{G}$ be in $\mathit{QCoh}((\mathit{Sch}/S)_{fppf}, \mathcal{O})$. Write $\mathcal{F} = \mathcal{H}^ a$ with $\mathcal{H}$ quasi-coherent on $S$. By Lemma 35.8.10 we have

\begin{align*} \mathcal{F}\text{ finite locally free in fppf topology} & \Leftrightarrow \mathcal{H}\text{ finite locally free on }S \\ & \Leftrightarrow \mathcal{F}\text{ finite locally free in ├ętale topology} \\ & \Leftrightarrow \mathcal{H}\text{ finite locally free on }S \\ & \Leftrightarrow \mathcal{F}\text{ finite locally free in Zariski topology} \end{align*}

This explains the parenthetical statement of part (3). Now, if these equivalent conditions hold, then $\mathcal{H}$ is finite locally free. The construction of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G})$ in Modules on Sites, Section 18.27 depends only on $\mathcal{F}$ and $\mathcal{G}$ as presheaves of modules (only whether the output $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ is a sheaf depends on whether $\mathcal{F}$ and $\mathcal{G}$ are sheaves). $\square$


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