# The Stacks Project

## Tag 00UU

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 39110–39122 (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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