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

edit

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

edit

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

The  -algebra   is not étale, since  .

Properties

edit

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

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

References

edit
  • 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