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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CDR. Beware of the difference between the letter 'O' and the digit '0'.