The Stacks project

Lemma 35.6.1. Let $k'/k$ be a (finite) Galois extension with Galois group $G$. Let $X$ be a scheme over $k$. The category of quasi-coherent $\mathcal{O}_ X$-modules is equivalent to the category of systems $(\mathcal{F}, (\varphi _\sigma )_{\sigma \in G})$ where

  1. $\mathcal{F}$ is a quasi-coherent module on $X_{k'}$,

  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 $\sigma , \tau \in G$.

Here $f_\sigma = \text{id}_ X \times \mathop{\mathrm{Spec}}(\sigma ) : X_{k'} \to X_{k'}$.

Proof. As seen above a datum $(\mathcal{F}, (\varphi _\sigma )_{\sigma \in G})$ as in the lemma is the same thing as a descent datum for the fpqc covering $\{ X_{k'} \to X\} $. Thus the lemma follows from Proposition 35.5.2. $\square$

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