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Tag 05CK

Chapter 10: Commutative Algebra > Section 10.81: Universally injective module maps

Lemma 10.81.11. Let $R \to S$ be a faithfully flat ring map. Then $R \to S$ is universally injective as a map of $R$-modules. In particular $R \cap IS = I$ for any ideal $I \subset R$.

Proof. Let $N$ be an $R$-module. We have to show that $N \to N \otimes_R S$ is injective. As $S$ is faithfully flat as an $R$-module, it suffices to prove this after tensoring with $S$. Hence it suffices to show that $N \otimes_R S \to N \otimes_R S \otimes_R S$, $n \otimes s \mapsto n \otimes 1 \otimes s$ is injective. This is true because there is a retraction, namely, $n \otimes s \otimes s' \mapsto n \otimes ss'$. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 19075–19080 (see updates for more information).

    \begin{lemma}
    \label{lemma-faithfully-flat-universally-injective}
    Let $R \to S$ be a faithfully flat ring map.
    Then $R \to S$ is universally injective as a map of $R$-modules.
    In particular $R \cap IS = I$ for any ideal $I \subset R$.
    \end{lemma}
    
    \begin{proof}
    Let $N$ be an $R$-module. We have to show that $N \to N \otimes_R S$ is
    injective. As $S$ is faithfully flat as an $R$-module, it suffices to prove
    this after tensoring with $S$. Hence it suffices to show that
    $N \otimes_R S \to N \otimes_R S \otimes_R S$,
    $n \otimes s \mapsto n \otimes 1 \otimes s$ is injective. This is true
    because there is a retraction, namely,
    $n \otimes s \otimes s' \mapsto n \otimes ss'$.
    \end{proof}

    Comments (3)

    Comment #1403 by Fred Rohrer (site) on April 13, 2015 a 9:07 pm UTC

    Replace the proof by "This is the same as Lemma 051A."

    Comment #1404 by jojo on April 14, 2015 a 5:51 am UTC

    I would say keep the proof, it's easier to read this way than having to go somewhere else to check it ;)

    Comment #1413 by Johan (site) on April 15, 2015 a 6:53 pm UTC

    OK, guys, I sent the first occurence of this lemma (051A) to the obsolete chapter. Thanks! See here.

    There are also 2 comments on Section 10.81: Commutative Algebra.

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