The Stacks project

Lemma 35.8.1. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf, \linebreak[0] fpqc\} $. The functor defined in ( satisfies the sheaf condition with respect to any $\tau $-covering $\{ T_ i \to T\} _{i \in I}$ of any scheme $T$ over $S$.

Proof. For $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf\} $ a $\tau $-covering is also a fpqc-covering, see the results in Topologies, Lemmas 34.4.2, 34.5.2, 34.6.2, 34.7.2, and 34.9.6. Hence it suffices to prove the theorem for a fpqc covering. Assume that $\{ f_ i : T_ i \to T\} _{i \in I}$ is an fpqc covering where $f : T \to S$ is given. Suppose that we have a family of sections $s_ i \in \Gamma (T_ i , f_ i^*f^*\mathcal{F})$ such that $s_ i|_{T_ i \times _ T T_ j} = s_ j|_{T_ i \times _ T T_ j}$. We have to find the correspond section $s \in \Gamma (T, f^*\mathcal{F})$. We can reinterpret the $s_ i$ as a family of maps $\varphi _ i : f_ i^*\mathcal{O}_ T = \mathcal{O}_{T_ i} \to f_ i^*f^*\mathcal{F}$ compatible with the canonical descent data associated to the quasi-coherent sheaves $\mathcal{O}_ T$ and $f^*\mathcal{F}$ on $T$. Hence by Proposition 35.5.2 we see that we may (uniquely) descend these to a map $\mathcal{O}_ T \to f^*\mathcal{F}$ which gives us our section $s$. $\square$

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