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

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