Lemma 35.6.2. Let X \to Y, G, and f_\sigma : X \to X be as above. The category of quasi-coherent \mathcal{O}_ Y-modules is equivalent to the category of systems (\mathcal{F}, (\varphi _\sigma )_{\sigma \in G}) where
\mathcal{F} is a quasi-coherent \mathcal{O}_ X-module,
\varphi _\sigma : \mathcal{F} \to f_\sigma ^*\mathcal{F} is an isomorphism of modules,
\varphi _{\sigma \tau } = f_\sigma ^*\varphi _\tau \circ \varphi _\sigma for all \sigma , \tau \in G.
Proof. Since X \to Y is surjective finite étale \{ X \to Y\} is an fpqc covering. Since G \times X \to X \times _ Y X, (\sigma , x) \mapsto (x, f_\sigma (x)) is an isomorphism, we see that G \times G \times X \to X \times _ Y X \times _ Y X, (\sigma , \tau , x) \mapsto (x, f_\sigma (x), f_{\sigma \tau }(x)) is an isomorphism too. Using these identifications, the category of data as in the lemma is the same as the category of descent data for quasi-coherent sheaves for the covering \{ x \to Y\} . Thus the lemma follows from Proposition 35.5.2. \square
Comments (1)
Comment #4085 by Jeroen van der Meer on