The Stacks project

Remarks 59.16.8. The results on descent of modules have several applications:

  1. The exactness of the Čech complex in positive degrees for the covering $\{ \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)\} $ where $A \to B$ is faithfully flat. This will give some vanishing of cohomology.

  2. If $(N, \varphi )$ is a descent datum with respect to a faithfully flat map $A \to B$, then the corresponding $A$-module is given by

    \[ M = \mathop{\mathrm{Ker}}\left( \begin{matrix} N & \longrightarrow & B \otimes _ A N \\ n & \longmapsto & 1 \otimes n - \varphi (n \otimes 1) \end{matrix} \right). \]

    See Descent, Proposition 35.3.9.


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Your email address will not be published.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