The Stacks project

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

  1. $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module,

  2. $\varphi _\sigma : \mathcal{F} \to f_\sigma ^*\mathcal{F}$ is an isomorphism of modules,

  3. $\varphi _{\sigma \tau } = f_\sigma ^*\varphi _\tau \circ \varphi _\sigma $ for all $\sigma , \tau \in G$.

Proof. Since $X \to Y$ is surjective finite ├ętale $\{ 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

Typo: In the last sentence of the proof, should be .

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