The Stacks project

Proposition 35.8.9. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} $.

  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 $\tau = Zariski$ or $\tau = {\acute{e}tale}$. 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 35.8.7 that the functor is well defined. By Lemma 35.8.8 the functor is fully faithful. 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 35.5.2 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)_\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 35.5.2) 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. $\square$

Comments (2)

Comment #5563 by Manuel Hoff on

I think there is a small typo in the first sentence of the third paragraph of the proof: should be replaced with .

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  • 1 comment(s) on Section 35.8: Quasi-coherent sheaves and topologies, I

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