# The Stacks Project

## Tag 023G

Definition 34.3.1. 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 a 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.

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 270–306 (see updates for more information).

\begin{definition}
\label{definition-descent-datum-modules}
Let $R \to A$ be a ring map.
\begin{enumerate}
\item A {\it descent datum $(N, \varphi)$ for modules
with respect to $R \to A$}
is given by an $A$-module $N$ and a isomorphism of
$A \otimes_R A$-modules
$$\varphi : N \otimes_R A \to A \otimes_R N$$
such that the {\it 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).
\item A {\it 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.
\end{enumerate}
\end{definition}

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