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 Template:Mvar be a field. Let Template:Mvar be a commutative unital associative Template:Mvar-algebra. Then Template:Mvar is called an étale Template:Mvar-algebra if any one of the following equivalent conditions holds: Template:Bulleted list

Examples

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

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

Properties

Let Template:Mvar denote the absolute Galois group of Template:Mvar. Then the category of étale Template:Mvar-algebras is equivalent to the category of finite [[group action|Template:Mvar-sets]] with continuous Template:Mvar-action. In particular, étale algebras of dimension Template:Mvar are classified by conjugacy classes of continuous group homomorphisms from Template:Mvar to the symmetric group Template:Math. These globalize to e.g. the definition of étale fundamental groups and categorify to Grothendieck's Galois theory.

References

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• Template:Citation http://www.jmilne.org/math/CourseNotes/FT.pdf