“Étale algebra”的意思、由来-开放百科全书

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Étale algebra

释义

Definitions
Examples
Properties
References

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 be a field and let be a -algebra. Then is called étale or separable if as -algebras, where is an algebraically closed extension of and is an integer {{harv|Bourbaki|1990|loc=page A.V.28-30}}.

Equivalently, is étale if it is isomorphic to a finite product of separable extensions of . When these extensions are all of finite degree, is said to be finite étale; in this case one can replace with a finite separable extension of 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 is an étale morphism.

Examples

Consider the -algebra . This is etale because it is a separable field extension.

A simple non-example is given by since .

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

{{citation|MR=1080964

|last=Bourbaki|first= N.
|title=Algebra. II. Chapters 4–7.
|series= Elements of Mathematics |place=Berlin|publisher= Springer-Verlag|year= 1990| isbn=3-540-19375-8 }}

{{citation|last=Milne|first=James|title=Field Theory}} http://www.jmilne.org/math/CourseNotes/FT.pdf

1 : Commutative algebra

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