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. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 35.5: Fpqc descent of quasi-coherent sheaves

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Your email address will not be published.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 023S. Beware of the difference between the letter 'O' and the digit '0'.