Definition 10.148.1. Let $R \to S$ be a ring map.

1. We say $R \to S$ is unramified if $R \to S$ is of finite type and $\Omega _{S/R} = 0$.

2. We say $R \to S$ is G-unramified if $R \to S$ is of finite presentation and $\Omega _{S/R} = 0$.

3. Given a prime $\mathfrak q$ of $S$ we say that $S$ is unramified at $\mathfrak q$ if there exists a $g \in S$, $g \not\in \mathfrak q$ such that $R \to S_ g$ is unramified.

4. Given a prime $\mathfrak q$ of $S$ we say that $S$ is G-unramified at $\mathfrak q$ if there exists a $g \in S$, $g \not\in \mathfrak q$ such that $R \to S_ g$ is G-unramified.

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