The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Proposition 34.8.11. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $.

  1. The functor $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence of categories

    \[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}((\mathit{Sch}/S)_\tau , \mathcal{O}) \]

    between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\mathcal{O}$-modules on the big $\tau $ site of $S$.

  2. Let $\tau = {\acute{e}tale}$, or $\tau = Zariski$. The functor $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence of categories

    \[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \mathit{QCoh}(S_\tau , \mathcal{O}) \]

    between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\mathcal{O}$-modules on the small $\tau $ site of $S$.

Proof. We have seen in Lemma 34.8.7 that the functor is well defined. It is straightforward to show that the functor is fully faithful (we omit the verification). To finish the proof we will show that a quasi-coherent $\mathcal{O}$-module on $(\mathit{Sch}/S)_\tau $ gives rise to a descent datum for quasi-coherent sheaves relative to a $\tau $-covering of $S$. Having produced this descent datum we will appeal to Proposition 34.5.2 to get the corresponding quasi-coherent sheaf on $S$.

Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}$-modules on the big $\tau $ site of $S$. By Modules on Sites, Definition 18.23.1 there exists a $\tau $-covering $\{ S_ i \to S\} _{i \in I}$ of $S$ such that each of the restrictions $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau }$ has a global presentation

\[ \bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow 0 \]

for some index sets $J_ i$ and $K_ i$. We claim that this implies that $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau }$ is $\mathcal{F}_ i^ a$ for some quasi-coherent sheaf $\mathcal{F}_ i$ on $S_ i$. Namely, this is clear for the direct sums $\bigoplus \nolimits _{k \in K_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau }$ and $\bigoplus \nolimits _{j \in J_ i} \mathcal{O}|_{(\mathit{Sch}/S_ i)_\tau }$. Hence we see that $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau }$ is a cokernel of a map $\varphi : \mathcal{K}_ i^ a \to \mathcal{L}_ i^ a$ for some quasi-coherent sheaves $\mathcal{K}_ i$, $\mathcal{L}_ i$ on $S_ i$. By the fully faithfulness of $(\ )^ a$ we see that $\varphi = \phi ^ a$ for some map of quasi-coherent sheaves $\phi : \mathcal{K}_ i \to \mathcal{L}_ i$ on $S_ i$. Then it is clear that $\mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau } \cong \mathop{\mathrm{Coker}}(\phi )^ a$ as claimed.

Since $\mathcal{G}$ lives on all of the category $(\mathit{Sch}/S_ i)_\tau $ we see that

\[ (\text{pr}_0^*\mathcal{F}_ i)^ a \cong \mathcal{G}|_{(\mathit{Sch}/(S_ i \times _ S S_ j))_\tau } \cong (\text{pr}_1^*\mathcal{F})^ a \]

as $\mathcal{O}$-modules on $(\mathit{Sch}/(S_ i \times _ S S_ j))_\tau $. Hence, using fully faithfulness again we get canonical isomorphisms

\[ \phi _{ij} : \text{pr}_0^*\mathcal{F}_ i \longrightarrow \text{pr}_1^*\mathcal{F}_ j \]

of quasi-coherent modules over $S_ i \times _ S S_ j$. We omit the verification that these satisfy the cocycle condition. Since they do we see by effectivity of descent for quasi-coherent sheaves and the covering $\{ S_ i \to S\} $ (Proposition 34.5.2) that there exists a quasi-coherent sheaf $\mathcal{F}$ on $S$ with $\mathcal{F}|_{S_ i} \cong \mathcal{F}_ i$ compatible with the given descent data. In other words we are given $\mathcal{O}$-module isomorphisms

\[ \phi _ i : \mathcal{F}^ a|_{(\mathit{Sch}/S_ i)_\tau } \longrightarrow \mathcal{G}|_{(\mathit{Sch}/S_ i)_\tau } \]

which agree over $S_ i \times _ S S_ j$. Hence, since $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}^ a, \mathcal{G})$ is a sheaf (Modules on Sites, Lemma 18.27.1), we conclude that there is a morphism of $\mathcal{O}$-modules $\mathcal{F}^ a \to \mathcal{G}$ recovering the isomorphisms $\phi _ i$ above. Hence this is an isomorphism and we win.

The case of the sites $S_{\acute{e}tale}$ and $S_{Zar}$ is proved in the exact same manner. $\square$


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