The Stacks project

Definition 74.3.3. Let $S$ be a scheme. Let $\{ U_ i \to U\} _{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target.

  1. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ U$-module. We call the unique descent on $\mathcal{F}$ datum with respect to the covering $\{ U \to U\} $ the trivial descent datum.

  2. The pullback of the trivial descent datum to $\{ U_ i \to U\} $ is called the canonical descent datum. Notation: $(\mathcal{F}|_{U_ i}, can)$.

  3. A descent datum $(\mathcal{F}_ i, \varphi _{ij})$ for quasi-coherent sheaves with respect to the given family is said to be effective if there exists a quasi-coherent sheaf $\mathcal{F}$ on $U$ such that $(\mathcal{F}_ i, \varphi _{ij})$ is isomorphic to $(\mathcal{F}|_{U_ i}, can)$.

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Your email address will not be published.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 04W5. Beware of the difference between the letter 'O' and the digit '0'.



View Definition 74.3.3 as pdf