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.

Not to be confused with a separable algebra.

Definitions

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

Equivalently, ${\displaystyle L}$ is étale if it is isomorphic to a finite product of separable extensions of ${\displaystyle K}$. When these extensions are all of finite degree, ${\displaystyle L}$ is said to be finite étale; in this case one can replace ${\displaystyle \Omega }$ with a finite separable extension of ${\displaystyle K}$ 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 ${\displaystyle \mathrm {Spec} \,L\to \mathrm {Spec} \,K}$ is an étale morphism.

Examples

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

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

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