Étale morphism of schemes

Let $f:X\to S$ be a morphism of schemes and let $x\in X$; put $s=f(x)$.

The morphism $f$ is étale at $x$ if it is smooth at $x$ (Smooth morphism of schemes) and has relative dimension $0$ at $x$ (Relative dimension of a smooth morphism at a point). Spelling out the two definitions, $f$ is étale at $x$ if and only if

1. $f$ is locally of finite presentation at $x$ (Locally finite presentation morphisms);
2. $f$ is flat at $x$ (Flat morphism of schemes);
3. for every field extension $K\mathrm{/}\kappa (s)$ the base-changed fibre $X_{s,K}=X_s{\times }_{\mathrm{Spec}\kappa (s)}\mathrm{Spec}K$ has a regular local ring (Geometrically regular algebras and geometrically regular fibres) at every point $y$ lying over the image of $x$, with ${\dim }_y{X}_{s,K}=0$ at each such point.

The morphism $f$ is étale if it is étale at every point of $X$. Thus an étale morphism is precisely a smooth morphism of pure relative dimension $0$; the number $0$ in the condition is the well-defined relative dimension of a smooth germ, so no additional choice is made and no separate well-definedness clause is needed. A morphism with empty source is étale vacuously, and the clause about points of a geometric fibre is vacuous for a fibre that is empty.

Étaleness is a condition on the germ of $f$ at $x$: shrinking the source to an open neighbourhood of $x$ or the target to an open neighbourhood of $s$ replaces the geometric fibre of the morphism by an open part of the geometric fibre of $f$ at $x$, and the local dimension at a point is unchanged by passing to the open part, so the conditions are preserved in both directions. The definition is not symmetric in $X$ and $S$ and gives no flatness or smoothness of $\mathrm{Spec}$ of a field extension; only the geometric fibres are constrained.

Relative dimension zero is the case of the definition in which the geometric fibres have no positive-dimensional local pieces, and the standard reformulation of étaleness as flatness together with unramifiedness (equivalently, the local standard form, and the equivalent description by an invertible Jacobian determinant with no free parameters) is proved later on this page; those equivalences are not part of this definition. In particular a localisation $S_g\to S$ and an open immersion are the basic examples, while the relative Frobenius of an affine line in positive characteristic is a surjective finite flat map that fails to be étale because its geometric fibres are nonreduced, as the examples page records. Degree greater than one alone does not obstruct étaleness: the disjoint union of two copies of $\mathrm{Spec}k$ over $\mathrm{Spec}k$ is étale of degree two, since on each open component the morphism is the identity.