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

1. 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.

2. We say that $f$ is étale if it is étale at every point of $X$.

3. 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 $R[x]_ h/(g)$.

Comment #2808 by Jonathan Gruner on

(3) is confusing because f is the name of the morphism and a polynomial.

Comment #2809 by Jonathan Gruner on

PS: Especially because morphisms and polynomials can both be monic!

There are also:

• 3 comment(s) on Section 29.36: Étale morphisms

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).