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 .
![{\displaystyle \mathbb {R} [x]/(x^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/561ba4f313353bce26f1a8ad04d45fd27924fcf1)
![{\displaystyle \mathbb {R} [x]/(x^{2})\otimes _{\mathbb {R} }\mathbb {C} \simeq \mathbb {C} [x]/(x^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64c62bd43455d8bff3dcddb836ef4603560fa959)
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, 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
This page was last edited on 10 May 2024, at 04:24