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Tag 00HB

Chapter 10: Commutative Algebra > Section 10.38: Flat modules and flat ring maps

Definition 10.38.1. Let $R$ be a ring.

  1. An $R$-module $M$ is called flat if whenever $N_1 \to N_2 \to N_3$ is an exact sequence of $R$-modules the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$ is exact as well.
  2. An $R$-module $M$ is called faithfully flat if the complex of $R$-modules $N_1 \to N_2 \to N_3$ is exact if and only if the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$ is exact.
  3. A ring map $R \to S$ is called flat if $S$ is flat as an $R$-module.
  4. A ring map $R \to S$ is called faithfully flat if $S$ is faithfully flat as an $R$-module.

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 8269–8287 (see updates for more information).

    \begin{definition}
    \label{definition-flat}
    Let $R$ be a ring.
    \begin{enumerate}
    \item An $R$-module $M$ is called {\it flat} if whenever
    $N_1 \to N_2 \to N_3$ is an exact sequence of $R$-modules
    the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$
    is exact as well.
    \item An $R$-module $M$ is called {\it faithfully flat} if the
    complex of $R$-modules
    $N_1 \to N_2 \to N_3$ is exact if and only if
    the sequence $M \otimes_R N_1 \to M \otimes_R N_2 \to M \otimes_R N_3$
    is exact.
    \item A ring map $R \to S$ is called {\it flat} if
    $S$ is flat as an $R$-module.
    \item A ring map $R \to S$ is called {\it faithfully flat} if
    $S$ is faithfully flat as an $R$-module.
    \end{enumerate}
    \end{definition}

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