Étale algebra explained

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 be a field. Let be a commutative unital associative -algebra. Then is called an étale -algebra if any one of the following equivalent conditions holds:

Examples

The

Q -algebra Q (i) is étale because it is a finite separable field extension.

The

R -algebra R [x]/(x ^{2)} is not étale, since R [x]/(x ^{2 ) ⊗} _{R C} \simeq C [x]/(x ^{2)} .

Properties

Let denote the absolute Galois group of . Then the category of étale -algebras is equivalent to the category of finite -sets with continuous -action. In particular, étale algebras of dimension are classified by conjugacy classes of continuous homomorphisms from to the symmetric group . These globalize to e.g. the definition of étale fundamental groups and categorify to Grothendieck's Galois theory.

References

• • http://www.jmilne.org/math/CourseNotes/FT.pdf