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]
- [math]\displaystyle{ L\otimes_{K} E\simeq E^n }[/math] for some field extension E of K and some nonnegative integer n.
- [math]\displaystyle{ L\otimes_{K} \overline{K} \simeq \overline{K}^n }[/math] for any algebraic closure [math]\displaystyle{ \overline{K} }[/math] 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 [math]\displaystyle{ \operatorname{Spec} L \to \operatorname{Spec} K }[/math] is an étale morphism.
Examples
The [math]\displaystyle{ \mathbb{Q} }[/math]-algebra [math]\displaystyle{ \mathbb{Q}(i) }[/math] is étale because it is a finite separable field extension.
The [math]\displaystyle{ \mathbb{R} }[/math]-algebra [math]\displaystyle{ \mathbb{R}[x]/(x^2) }[/math] is not étale, since [math]\displaystyle{ \mathbb{R}[x]/(x^2)\otimes_\mathbb{R}\mathbb{C} \simeq \mathbb{C}[x]/(x^2) }[/math].
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
- ↑ (Bourbaki 1990)
References
- Bourbaki, N. (1990), Algebra. II. Chapters 4–7., Elements of Mathematics, Berlin: Springer-Verlag, ISBN 3-540-19375-8
- Milne, James, Field Theory http://www.jmilne.org/math/CourseNotes/FT.pdf
 |  |
