Definition 29.36.1. Let $f : X \to S$ be a morphism of schemes.

We say that $f$ is

*étale at $x \in X$*if there exists an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset X$ of $x$ and affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is étale.We say that $f$ is

*étale*if it is étale at every point of $X$.A morphism of affine schemes $f : X \to S$ is called

*standard étale*if $X \to S$ is isomorphic to\[ \mathop{\mathrm{Spec}}(R[x]_ h/(g)) \to \mathop{\mathrm{Spec}}(R) \]where $R \to R[x]_ h/(g)$ is a standard étale ring map, see Algebra, Definition 10.144.1, i.e., $g$ is monic and $g'$ invertible in $R[x]_ h/(g)$.

## Comments (3)

Comment #2808 by Jonathan Gruner on

Comment #2809 by Jonathan Gruner on

Comment #2911 by Johan on

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