Definition 35.3.1. Let $R \to A$ be a ring map.

A

*descent datum $(N, \varphi )$ for modules with respect to $R \to A$*is given by an $A$-module $N$ and an isomorphism of $A \otimes _ R A$-modules\[ \varphi : N \otimes _ R A \to A \otimes _ R N \]such that the

*cocycle condition*holds: the diagram of $A \otimes _ R A \otimes _ R A$-module maps\[ \xymatrix{ N \otimes _ R A \otimes _ R A \ar[rr]_{\varphi _{02}} \ar[rd]_{\varphi _{01}} & & A \otimes _ R A \otimes _ R N \\ & A \otimes _ R N \otimes _ R A \ar[ru]_{\varphi _{12}} & } \]commutes (see below for notation).

A

*morphism $(N, \varphi ) \to (N', \varphi ')$ of descent data*is a morphism of $A$-modules $\psi : N \to N'$ such that the diagram\[ \xymatrix{ N \otimes _ R A \ar[r]_\varphi \ar[d]_{\psi \otimes \text{id}_ A} & A \otimes _ R N \ar[d]^{\text{id}_ A \otimes \psi } \\ N' \otimes _ R A \ar[r]^{\varphi '} & A \otimes _ R N' } \]is commutative.

## Comments (2)

Comment #3207 by Dario Weißmann on

Comment #3311 by Johan on

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