Étale morphism of rings

Summary

An étale morphism of rings is a map $R \to A$ of commutative rings that is both smooth and unramified. Equivalently, $A$ is finitely presented, flat and unramified over $R$. It is the algebraic analogue of a local diffeomorphism, or of a covering map. Over a field $k$, the étale algebras are the finite products of finite separable field extensions of $k$. Étale maps are the basis of étale cohomology and of the Henselization of a local ring.