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Tag 00UU

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

Lemma 10.147.2. Let $R \to S$ be a ring map. The following are equivalent

  1. $R \to S$ is formally unramified and of finite type, and
  2. $R \to S$ is unramified.

Moreover, also the following are equivalent

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

Proof. Follows from Lemma 10.144.2 and the definitions. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 39187–39199 (see updates for more information).

    \begin{lemma}
    \label{lemma-formally-unramified-unramified}
    Let $R \to S$ be a ring map. The following are equivalent
    \begin{enumerate}
    \item $R \to S$ is formally unramified and of finite type, and
    \item $R \to S$ is unramified.
    \end{enumerate}
    Moreover, also the following are equivalent
    \begin{enumerate}
    \item $R \to S$ is formally unramified and of finite presentation, and
    \item $R \to S$ is G-unramified.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Follows from Lemma \ref{lemma-characterize-formally-unramified}
    and the definitions.
    \end{proof}

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