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 $K$ be a field and ${\mathfrak {R}}$ be a $K$-algebra. Then ${\mathfrak {R}}$ is called étale or separable if ${\mathfrak {R}}\otimes _{K}{\bar {K}}\cong {\bar {K}}\times \cdots \times {\bar {K}}$ (Bourbaki 1990, page A.V.28).

Alternatively an étale algebra is an algebra isomorphic to a finite product of separable extensions.

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 algebra is étale if and only if $\mathrm {Spec} \,{\mathfrak {R}}\to \mathrm {Spec} \,K$ 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.

References

• Bourbaki, N. (1990), Algebra. II. Chapters 4–7., Elements of Mathematics, Berlin: Springer-Verlag, ISBN 3-540-19375-8, MR 1080964