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Tag 0584

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

Lemma 10.38.9. Let $R$ be a ring. Let $S \to S'$ be a faithfully flat map of $R$-algebras. Let $M$ be a module over $S$, and set $M' = S' \otimes_S M$. Then $M$ is flat over $R$ if and only if $M'$ is flat over $R$.

Proof. Let $N \to N'$ be an injection of $R$-modules. By the faithful flatness of $S \to S'$ we have $$ \mathop{\rm Ker}(N \otimes_R M \to N' \otimes_R M) \otimes_S S' = \mathop{\rm Ker}(N \otimes_R M' \to N' \otimes_R M') $$ Hence the equivalence of the lemma follows from the second characterization of flatness in Lemma 10.38.5. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 8599–8605 (see updates for more information).

    \begin{lemma}
    \label{lemma-flatness-descends-more-general}
    Let $R$ be a ring.
    Let $S \to S'$ be a faithfully flat map of $R$-algebras.
    Let $M$ be a module over $S$, and set $M' = S' \otimes_S M$.
    Then $M$ is flat over $R$ if and only if $M'$ is flat over $R$.
    \end{lemma}
    
    \begin{proof}
    Let $N \to N'$ be an injection of $R$-modules. By the faithful flatness
    of $S \to S'$ we have
    $$
    \Ker(N \otimes_R M \to N' \otimes_R M) \otimes_S S'
    =
    \Ker(N \otimes_R M' \to N' \otimes_R M')
    $$
    Hence the equivalence of the lemma follows from the second characterization
    of flatness in
    Lemma \ref{lemma-flat}.
    \end{proof}

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