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:[1]

  • for some field extension E of K and some nonnegative integer n.
  • for any algebraic closure 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 is an étale morphism.

Examples

The -algebra is étale because it is a finite separable field extension.

The -algebra is not étale, since .

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 homomorphisms from G to the symmetric group Sn. These globalize to e.g. the definition of étale fundamental groups and categorify to Grothendieck's Galois theory.

Notes

  1. (Bourbaki 1990, page A.V.28-30)

References

  • Bourbaki, N. (1990), Algebra. II. Chapters 4–7., Elements of Mathematics, Berlin: Springer-Verlag, ISBN 3-540-19375-8, MR 1080964
  • Milne, James, Field Theory http://www.jmilne.org/math/CourseNotes/FT.pdf


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