Étale algebra

Definitions

Let K be a field and let L be a K -algebra. Then L is called étale or separable if L ⊗ K Ω ≃ Ω n , where Ω is an algebraically closed extension of K and n ≥ 0 is an integer (Bourbaki 1990, page A.V.28-30).

Equivalently, L is étale if it is isomorphic to a finite product of separable extensions of K . When these extensions are all of finite degree, L is said to be finite étale; in this case one can replace Ω with a finite separable extension of K in the definition above.

A third definition says that an étale algebra is a finite dimensional commutative algebra whose trace form (x,y) = Tr(xy) is non-degenerate

The name "étale algebra" comes from the fact that a finite dimensional commutative algebra over a field is étale if and only if S p e c L → S p e c K is an étale morphism.

Properties

The category of étale algebras over a field k is equivalent to the category of finite G-sets (with continuous G-action), where G is the absolute Galois group of k. In particular étale algebras of dimension n are classified by conjugacy classes of continuous homomorphisms from the absolute Galois group to the symmetric group S_n.