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Tag 023G

Chapter 34: Descent > Section 34.3: Descent for modules

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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