Definition 59.16.5. Let $A \to B$ be a ring map and $N$ a $B$-module. A descent datum for $N$ with respect to $A \to B$ is an isomorphism $\varphi : N \otimes _ A B \cong B \otimes _ A N$ of $B \otimes _ A B$-modules such that the diagram of $B \otimes _ A B \otimes _ A B$-modules

$\xymatrix{ {N \otimes _ A B \otimes _ A B} \ar[dr]_{\varphi _{02}} \ar[rr]^{\varphi _{01}} & & {B \otimes _ A N \otimes _ A B} \ar[dl]^{\varphi _{12}} \\ & {B \otimes _ A B \otimes _ A N} }$

commutes where $\varphi _{01} = \varphi \otimes \text{id}_ B$ and similarly for $\varphi _{12}$ and $\varphi _{02}$.

Comment #1702 by Yogesh More on

Minor remark: The $\phi_{ij}$ are of course different from the $\phi_{ij}$ in definition 49.16.1; here you mean presumably $\phi_{ij}=pr^*_{ij}\phi$, and then are you sure the arrows are labeled correctly in the commutative diagram? I would have guessed it would be $\phi_{02}=\phi_{12} \circ \phi_{01}$.

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