# The Stacks Project

## Tag 02G4

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

1. 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.
2. 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.
3. We say that $f$ is unramified if it is unramified at every point of $X$.
4. We say that $f$ is G-unramified if 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 6956–6975 (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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