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.

Definitions

Let K be a field. 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]

• ${\displaystyle L\otimes _{K}E\simeq E^{n}}$ for some field extension E of K and some nonnegative integer n.
• ${\displaystyle L\otimes _{K}{\overline {K}}\simeq {\overline {K}}^{n}}$ for any algebraic closure ${\displaystyle {\overline {K}}}$ of K and some nonnegative integer n.
• L is isomorphic to a finite product of finite separable field extensions of K.
• L is finite-dimensional over K, and the trace form Tr(xy) is nondegenerate.
• The morphism of schemes ${\displaystyle \operatorname {Spec} L\to \operatorname {Spec} K}$ is an étale morphism.

Examples

The ${\displaystyle \mathbb {Q} }$-algebra ${\displaystyle \mathbb {Q} (i)}$ is étale because it is a finite separable field extension.

The ${\displaystyle \mathbb {R} }$-algebra ${\displaystyle \mathbb {R} [x]/(x^{2})}$ of dual numbers is not étale, since ${\displaystyle \mathbb {R} [x]/(x^{2})\otimes _{\mathbb {R} }\mathbb {C} \simeq \mathbb {C} [x]/(x^{2})}$.

Properties

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 group homomorphisms from G to the symmetric group S_n. These globalize to e.g. the definition of étale fundamental groups and categorify to Grothendieck's Galois theory.