Étale algebra

In commutative algebra, an étale or separable algebra is a special type of algebra, one that is isomorphic to a finite product of separable extensions.

Definitions

Let $K$ be a field and let $L$ be a $K$-algebra. Then $L$ is called étale or separable if $L\otimes _{K}\Omega \simeq \Omega ^{n}$ as $K$-algebras, where $\Omega$ is an algebraically closed extension of $K$ and $n\geq 0$ is an integer (Bourbaki 1990, page A.V.28-30).

Equivalently, $L$ is étale if it is isomorphic to a finite product of separable extensions of $K$. When these extensions are all of finite degree, $L$ is said to be finite étale; in this case one can replace $\Omega$ with a finite separable extension of $K$ in the definition above.

A third definition says that an étale algebra is a finite dimensional commutative algebra whose trace form (x,y) = Tr(xy) is non-degenerate.

The name "étale algebra" comes from the fact that a finite dimensional commutative algebra over a field is étale if and only if $\mathrm {Spec} \,L\to \mathrm {Spec} \,K$ is an étale morphism.

Examples

Consider the $\mathbb {Q}$-algebra $\mathbb {Q} (i)$. This is etale because it is a separable field extension.

A simple non-example is given by $\mathbb {R} [x]/(x^{2})$ since $\mathbb {R} [x]/(x^{2})\otimes _{\mathbb {R} }\mathbb {C} \cong \mathbb {C} [x]/(x^{2})$.

Properties

The category of étale algebras over a field k is equivalent to the category of finite G-sets (with continuous G-action), where G is the absolute Galois group of k. In particular étale algebras of dimension n are classified by conjugacy classes of continuous homomorphisms from the absolute Galois group to the symmetric group S_n.

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