## 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

- $R \to S$ is formally unramified and of finite type, and
- $R \to S$ is unramified.
Moreover, also the following are equivalent

- $R \to S$ is formally unramified and of finite presentation, and
- $R \to S$ 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 39112–39124 (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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