The Stacks project

Lemma 73.22.8. Let $S$ be a scheme. Let $\mathcal{U}' = \{ X'_ i \to X'\} _{i \in I'}$ and $\mathcal{U} = \{ X_ j \to X\} _{i \in I}$ be families of morphisms with fixed target. Let $\alpha : I' \to I$, $g : X' \to X$ and $g_ i : X'_ i \to X_{\alpha (i)}$ be a morphism of families of maps with fixed target, see Sites, Definition 7.8.1.

  1. Let $(V_ i, \varphi _{ij})$ be a descent datum relative to the family $\mathcal{U}$. The system

    \[ \left( g_ i^*V_{\alpha (i)}, (g_ i \times g_ j)^*\varphi _{\alpha (i) \alpha (j)} \right) \]

    (with notation as in Remark 73.22.4) is a descent datum relative to $\mathcal{U}'$.

  2. This construction defines a functor between the category of descent data relative to $\mathcal{U}$ and the category of descent data relative to $\mathcal{U}'$.

  3. Given a second $\beta : I' \to I$, $h : X' \to X$ and $h'_ i : X'_ i \to X_{\beta (i)}$ morphism of families of maps with fixed target, then if $g = h$ the two resulting functors between descent data are canonically isomorphic.

  4. These functors agree, via Lemma 73.22.5, with the pullback functors constructed in Lemma 73.22.6.

Proof. This follows from Lemma 73.22.6 via the correspondence of Lemma 73.22.5. $\square$


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Your email address will not be published.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