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:

• $L{\otimes }_{K}E\simeq E^n$ for some field extension E of K and some nonnegative integer n.
• $L{\otimes }_K\bar{K}\simeq \stackrel{n}{\stackrel{‾}{K}}$ for any algebraic closure $\bar{K}$ 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 $\mathrm{Spec}L\to \mathrm{Spec}K$ is an étale morphism.

Examples

The $\mathbb{\mathbb{Q}}$-algebra $\mathbb{\mathbb{Q}}(i)$ 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.

References

• Bourbaki, N. (1990), Algebra. II. Chapters 4–7., Elements of Mathematics, Berlin: Springer-Verlag, ISBN 3-540-19375-8 
• Milne, James, Field Theory  http://www.jmilne.org/math/CourseNotes/FT.pdf