The Stacks project

Remark 74.22.2. Let $S$ be a scheme. Let $Y \to X$ be a morphism of algebraic spaces over $S$. Let $(V/Y, \varphi )$ be a descent datum relative to $Y \to X$. We may think of the isomorphism $\varphi $ as an isomorphism

\[ (Y \times _ X Y) \times _{\text{pr}_0, Y} V \longrightarrow (Y \times _ X Y) \times _{\text{pr}_1, Y} V \]

of algebraic spaces over $Y \times _ X Y$. So loosely speaking one may think of $\varphi $ as a map $\varphi : \text{pr}_0^*V \to \text{pr}_1^*V$1. The cocycle condition then says that $\text{pr}_{02}^*\varphi = \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi $. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.

[1] Unfortunately, we have chosen the “wrong” direction for our arrow here. In Definitions 74.22.1 and 74.22.3 we should have the opposite direction to what was done in Definition 74.3.1 by the general principle that “functions” and “spaces” are dual.

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Your email address will not be published.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