Definition 10.151.1. Let $R \to S$ be a ring map.

We say $R \to S$ is

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

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