The Stacks project

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

1. 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).

2. 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.