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 separable field extensions.

An étale algebra is a special sort of commutative separable algebra.

Let L be a commutative unital associative K-algebra.

Then L is called an étale K-algebra if any one of the following equivalent conditions holds:[1] The

is étale because it is a finite separable field extension.

Let G denote the absolute Galois group of K. Then the category of étale K-algebras is equivalent to the category of finite G-sets with continuous G-action.

In particular, étale algebras of dimension n are classified by conjugacy classes of continuous homomorphisms from G to the symmetric group Sn.

These globalize to e.g. the definition of étale fundamental groups and categorify to Grothendieck's Galois theory.