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 34.9.9. 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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