Étale algebra

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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]

  • L ⊗ K E ≃ E n {\displaystyle L\otimes _{K}E\simeq E^{n}} {\displaystyle L\otimes _{K}E\simeq E^{n}} for some field extension E of K and some nonnegative integer n.
  • L ⊗ K K ¯ ≃ K ¯ n {\displaystyle L\otimes _{K}{\overline {K}}\simeq {\overline {K}}^{n}} {\displaystyle L\otimes _{K}{\overline {K}}\simeq {\overline {K}}^{n}} for any algebraic closure K ¯ {\displaystyle {\overline {K}}} {\displaystyle {\overline {K}}} 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 Spec ⁡ L → Spec ⁡ K {\displaystyle \operatorname {Spec} L\to \operatorname {Spec} K} {\displaystyle \operatorname {Spec} L\to \operatorname {Spec} K} is an étale morphism.

Examples

The Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} }-algebra Q ( i ) {\displaystyle \mathbb {Q} (i)} {\displaystyle \mathbb {Q} (i)} is étale because it is a finite separable field extension.

The R {\displaystyle \mathbb {R} } {\displaystyle \mathbb {R} }-algebra R [ x ] / ( x 2 ) {\displaystyle \mathbb {R} [x]/(x^{2})} {\displaystyle \mathbb {R} [x]/(x^{2})} of dual numbers is not étale, since R [ x ] / ( x 2 ) ⊗ R C ≃ C [ x ] / ( x 2 ) {\displaystyle \mathbb {R} [x]/(x^{2})\otimes _{\mathbb {R} }\mathbb {C} \simeq \mathbb {C} [x]/(x^{2})} {\displaystyle \mathbb {R} [x]/(x^{2})\otimes _{\mathbb {R} }\mathbb {C} \simeq \mathbb {C} [x]/(x^{2})}.

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 group 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