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]

$L\otimes _{K}E\simeq E^{n}$ for some field extension $E$ of $K$ and some nonnegative integer $n$ .
$L\otimes _{K}{\overline {K}}\simeq {\overline {K}}^{n}$ for any algebraic closure ${\overline {K}}$ 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 $\operatorname {Spec} L\to \operatorname {Spec} K$ is an étale morphism.

Examples

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

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

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 homomorphisms from $G$ to the symmetric group $S n$ . 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

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[1] (Bourbaki 1990, page A.V.28-30)

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