Étale algebra
In commutative algebra, an étale or separable algebra is a special type of algebra, one that is isomorphic to a finite product of separable extensions.
Definitions
Let be a field and let be a -algebra. Then is called étale or separable if , where is an algebraically closed extension of and is an integer (Bourbaki 1990, page A.V.28-30).
Equivalently, is étale if it is isomorphic to a finite product of separable extensions of . When these extensions are all of finite degree, is said to be finite étale; in this case one can replace with a finite separable extension of 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 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 Sn.
References
- Bourbaki, N. (1990), Algebra. II. Chapters 4–7., Elements of Mathematics, Berlin: Springer-Verlag, ISBN 3-540-19375-8, MR 1080964