Étale algebra

In

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

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,
MR 1080964

Milne, James, Field Theory http://www.jmilne.org/math/CourseNotes/FT.pdf

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

[1]