The Stacks project

Theorem 58.17.4 (Meta theorem on quasi-coherent sheaves). Let $S$ be a scheme. Let $\mathcal{C}$ be a site. Assume that

  1. the underlying category $\mathcal{C}$ is a full subcategory of $\mathit{Sch}/S$,

  2. any Zariski covering of $T \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ can be refined by a covering of $\mathcal{C}$,

  3. $S/S$ is an object of $\mathcal{C}$,

  4. every covering of $\mathcal{C}$ is an fpqc covering of schemes.

Then the presheaf $\mathcal{O}$ is a sheaf on $\mathcal{C}$ and any quasi-coherent $\mathcal{O}$-module on $(\mathcal{C}, \mathcal{O})$ is of the form $\mathcal{F}^ a$ for some quasi-coherent sheaf $\mathcal{F}$ on $S$.

Proof. After some formal arguments this is exactly Theorem 58.16.2. Details omitted. In Descent, Proposition 35.8.11 we prove a more precise version of the theorem for the big Zariski, fppf, étale, smooth, and syntomic sites of $S$, as well as the small Zariski and étale sites of $S$. $\square$


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