Finite étale algebra

Let $R$ be a commutative ring. A commutative $R$ -algebra $A$ is a finite étale $R$ -algebra if the induced morphism

\operatorname{Spec}A\longrightarrow\operatorname{Spec}R

is finite and étale. Equivalently, $A$ is a finitely generated projective $R$ -module and is unramified over $R$ ; the latter can be expressed as

\Omega^1_{A/R}=0.

Classification over a field

When $R=F$ is a field, every finite étale algebra has the form

A\cong K_1\times\cdots\times K_r

for finite separable extensions $K_i/F$ . It is a field exactly when its spectrum is connected.

Remarks

Étale morphisms are algebraic analogues of local diffeomorphisms, but the notions are not identical. A general scheme has no underlying classical smooth manifold, and in positive characteristic separability is indispensable.