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Tag 039X

Chapter 34: Descent > Section 34.3: Descent for modules

Lemma 34.3.8. Let $R \to A$ be a faithfully flat ring map, and let $R \to R'$ be faithfully flat. Set $A' = R' \otimes_R A$. If all descent data for $R' \to A'$ are effective, then so are all descent data for $R \to A$.

Proof. Let $(N, \varphi)$ be a descent datum for $R \to A$. Set $N' = R' \otimes_R N = A' \otimes_A N$, and denote $\varphi' = \text{id}_{R'} \otimes \varphi$ the base change of the descent datum $\varphi$. Then $(N', \varphi')$ is a descent datum for $R' \to A'$ and $H^0(s(N'_\bullet)) = R' \otimes_R H^0(s(N_\bullet))$. Moreover, the map $A' \otimes_{R'} H^0(s(N'_\bullet)) \to N'$ is identified with the base change of the $A$-module map $A \otimes_R H^0(s(N)) \to N$ via the faithfully flat map $A \to A'$. Hence we conclude by Lemma 34.3.7. $\square$

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 615–620 (see updates for more information).

    \begin{lemma}
    \label{lemma-descent-descends}
    Let $R \to A$ be a faithfully flat ring map, and let $R \to R'$
    be faithfully flat. Set $A' = R' \otimes_R A$. If all descent data
    for $R' \to A'$ are effective, then so are all descent data for $R \to A$.
    \end{lemma}
    
    \begin{proof}
    Let $(N, \varphi)$ be a descent datum for $R \to A$.
    Set $N' = R' \otimes_R N = A' \otimes_A N$, and denote
    $\varphi' = \text{id}_{R'} \otimes \varphi$ the base change
    of the descent datum $\varphi$. Then $(N', \varphi')$ is
    a descent datum for $R' \to A'$ and
    $H^0(s(N'_\bullet)) = R' \otimes_R H^0(s(N_\bullet))$.
    Moreover, the map
    $A' \otimes_{R'} H^0(s(N'_\bullet)) \to N'$ is identified
    with the base change of the $A$-module map
    $A \otimes_R H^0(s(N)) \to N$ via the faithfully flat map
    $A \to A'$. Hence we conclude by Lemma \ref{lemma-recognize-effective}.
    \end{proof}

    Comments (3)

    Comment #2874 by Ko Aoki on October 5, 2017 a 3:57 pm UTC

    Possibly wrong statement: The proof uses 039W, but it holds under an additional condition that $R \to A$ is faithfully flat. So I think faithfully flatness of $R \to A$ is needed. The same for 039Y.

    Comment #2936 by Johan (site) on October 10, 2017 a 2:15 am UTC

    Thanks, fixed here.

    Comment #2952 by Ko Aoki on October 11, 2017 a 12:50 pm UTC

    Thank you for fixing. (1) of 039Y contains the same error.

    There are also 2 comments on Section 34.3: Descent.

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