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The Stacks project

Lemma 35.5.1. Let S be an affine scheme. Let \mathcal{U} = \{ f_ i : U_ i \to S\} _{i = 1, \ldots , n} be a standard fpqc covering of S, see Topologies, Definition 34.9.10. Any descent datum on quasi-coherent sheaves for \mathcal{U} = \{ U_ i \to S\} is effective. Moreover, the functor from the category of quasi-coherent \mathcal{O}_ S-modules to the category of descent data with respect to \mathcal{U} is fully faithful.

Proof. This is a restatement of Proposition 35.3.9 in terms of schemes. First, note that a descent datum \xi for quasi-coherent sheaves with respect to \mathcal{U} is exactly the same as a descent datum \xi ' for quasi-coherent sheaves with respect to the covering \mathcal{U}' = \{ \coprod _{i = 1, \ldots , n} U_ i \to S\} . Moreover, effectivity for \xi is the same as effectivity for \xi '. Hence we may assume n = 1, i.e., \mathcal{U} = \{ U \to S\} where U and S are affine. In this case descent data correspond to descent data on modules with respect to the ring map

\Gamma (S, \mathcal{O}) \longrightarrow \Gamma (U, \mathcal{O}).

Since U \to S is surjective and flat, we see that this ring map is faithfully flat. In other words, Proposition 35.3.9 applies and we win. \square


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