The Stacks project

Definition 59.16.1. Let $\mathcal{U} = \{ t_ i : T_ i \to T\} _{i \in I}$ be a family of morphisms of schemes with fixed target. A descent datum for quasi-coherent sheaves with respect to $\mathcal{U}$ is a collection $((\mathcal{F}_ i)_{i \in I}, (\varphi _{ij})_{i, j \in I})$ where

  1. $\mathcal{F}_ i$ is a quasi-coherent sheaf on $T_ i$, and

  2. $\varphi _{ij} : \text{pr}_0^* \mathcal{F}_ i \to \text{pr}_1^* \mathcal{F}_ j$ is an isomorphism of modules on $T_ i \times _ T T_ j$,

such that the cocycle condition holds: the diagrams

\[ \xymatrix{ \text{pr}_0^*\mathcal{F}_ i \ar[dr]_{\text{pr}_{02}^*\varphi _{ik}} \ar[rr]^{\text{pr}_{01}^*\varphi _{ij}} & & \text{pr}_1^*\mathcal{F}_ j \ar[dl]^{\text{pr}_{12}^*\varphi _{jk}} \\ & \text{pr}_2^*\mathcal{F}_ k } \]

commute on $T_ i \times _ T T_ j \times _ T T_ k$. This descent datum is called effective if there exist a quasi-coherent sheaf $\mathcal{F}$ over $T$ and $\mathcal{O}_{T_ i}$-module isomorphisms $\varphi _ i : t_ i^* \mathcal{F} \cong \mathcal{F}_ i$ compatible with the maps $\varphi _{ij}$, namely

\[ \varphi _{ij} = \text{pr}_1^* (\varphi _ j) \circ \text{pr}_0^* (\varphi _ i)^{-1}. \]

Comments (0)

There are also:

  • 2 comment(s) on Section 59.16: Faithfully flat descent

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Your email address will not be published.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 03O7. Beware of the difference between the letter 'O' and the digit '0'.