Étale algebra

In commutative algebra, an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite field extension">field (mathematics)">field extensions. An étale algebra is a special sort of commutative separable algebra.

Definitions

Let be a field. Let be a commutative unital associative -algebra. Then is called an étale -algebra if any one of the following equivalent conditions holds:

Examples

The -algebra is étale because it is a finite separable field extension.
The -algebra of dual numbers is not étale, since.

Properties

Let denote the absolute [Galois group] of. Then the category of étale -algebras is equivalent to the category of finite -sets with continuous -action. In particular, étale algebras of dimension are classified by conjugacy classes of continuous group homomorphisms from to the symmetric group. These globalize to e.g. the definition of étale fundamental groups and categorify to Grothendieck's Galois theory.