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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.

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