Étale 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: |L\otimes_{K} E\simeq E^n for some field extension E of K and some nonnegative integer n. |L\otimes_{K} \overline{K} \simeq \overline{K}^n for any algebraic closure \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 \operatorname{Spec} L \to \operatorname{Spec} K is an étale morphism.

Examples

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

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

Notes

References

• {{citation|mr=1080964
• http://www.jmilne.org/math/CourseNotes/FT.pdf

References

1. {{harv. Bourbaki. 1990