## Tag `02G4`

Chapter 28: Morphisms of Schemes > Section 28.33: Unramified morphisms

Definition 28.33.1. Let $f : X \to S$ be a morphism of schemes.

- We say that $f$ is
unramified at $x \in X$if there exists an affine open neighbourhood $\mathop{\rm Spec}(A) = U \subset X$ of $x$ and affine open $\mathop{\rm Spec}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is unramified.- We say that $f$ is
G-unramified at $x \in X$if there exists an affine open neighbourhood $\mathop{\rm Spec}(A) = U \subset X$ of $x$ and affine open $\mathop{\rm Spec}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is G-unramified.- We say that $f$ is
unramifiedif it is unramified at every point of $X$.- We say that $f$ is
G-unramifiedif it is G-unramified at every point of $X$.

The code snippet corresponding to this tag is a part of the file `morphisms.tex` and is located in lines 6949–6968 (see updates for more information).

```
\begin{definition}
\label{definition-unramified}
Let $f : X \to S$ be a morphism of schemes.
\begin{enumerate}
\item We say that $f$ is {\it unramified at $x \in X$} if
there exists an affine open neighbourhood $\Spec(A) = U \subset X$
of $x$ and affine open $\Spec(R) = V \subset S$
with $f(U) \subset V$ such that the induced ring map
$R \to A$ is unramified.
\item We say that $f$ is {\it G-unramified at $x \in X$} if
there exists an affine open neighbourhood $\Spec(A) = U \subset X$
of $x$ and affine open $\Spec(R) = V \subset S$
with $f(U) \subset V$ such that the induced ring map
$R \to A$ is G-unramified.
\item We say that $f$ is {\it unramified} if it is unramified
at every point of $X$.
\item We say that $f$ is {\it G-unramified} if it is G-unramified
at every point of $X$.
\end{enumerate}
\end{definition}
```

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