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Tag 08WD

Chapter 10: Commutative Algebra > Section 10.147: Unramified ring maps

Lemma 10.147.8. Let $R \to S$ be a ring map. The following are equivalent

  1. $R \to S$ is étale,
  2. $R \to S$ is flat and G-unramified, and
  3. $R \to S$ is flat, unramified, and of finite presentation,

Proof. Parts (2) and (3) are equivalent by definition. The implication (1) $\Rightarrow$ (3) follows from the fact that étale ring maps are of finite presentation, Lemma 10.141.3 (flatness of étale maps), and Lemma 10.147.3 (étale maps are unramified). Conversely, the characterization of étale ring maps in Lemma 10.141.7 and the structure of unramified ring maps in Lemma 10.147.5 shows that (3) implies (1). (This uses that $R \to S$ is étale if $R \to S$ is étale at every prime $\mathfrak q \subset S$, see Lemma 10.141.3.) $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 39352–39360 (see updates for more information).

    \begin{lemma}
    \label{lemma-etale-flat-unramified-finite-presentation}
    Let $R \to S$ be a ring map. The following are equivalent
    \begin{enumerate}
    \item $R \to S$ is \'etale,
    \item $R \to S$ is flat and G-unramified, and
    \item $R \to S$ is flat, unramified, and of finite presentation,
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Parts (2) and (3) are equivalent by definition.
    The implication (1) $\Rightarrow$ (3) follows from
    the fact that \'etale ring maps are of finite presentation,
    Lemma \ref{lemma-etale} (flatness of \'etale maps), and
    Lemma \ref{lemma-unramified} (\'etale maps are unramified).
    Conversely, the characterization of \'etale ring maps in
    Lemma \ref{lemma-characterize-etale}
    and the structure of unramified ring maps in
    Lemma \ref{lemma-unramified-at-prime}
    shows that (3) implies (1). (This uses that $R \to S$
    is \'etale if $R \to S$ is \'etale at every prime $\mathfrak q \subset S$,
    see Lemma \ref{lemma-etale}.)
    \end{proof}

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