Étale algebra: Difference between revisions

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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 $K$ be a field. Let $L$ be a commutative unital 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, ISBN 3-540-19375-8, MR 1080964
Milne, James, Field Theory http://www.jmilne.org/math/CourseNotes/FT.pdf

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

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+In [[commutative algebra]], an '''étale algebra''' over a field is a special type of [[algebra (ring theory)|algebra]], one that is [[isomorphism|isomorphic]] to a finite product of finite separable field extensions.  An étale algebra is a special sort of commutative [[separable algebra]].
-{{unreferenced|date=September 2009}}
-In [[mathematics]], more specifically in [[algebra]], an '''étale''' or '''separable''' [[algebra (ring theory)|algebra]] is a special type of algebra.
-== Definition ==
+== Definitions ==
-Let <math>K</math> be a [[field (mathematics)|field]] and <math>\mathfrak{R}</math> be a <math>K</math>-algebra. Then <math>\mathfrak{R}</math> is called '''étale''' or '''[[separable algebra|separable]]''' if <math>\mathfrak{R}\otimes _{K}\bar{K}\cong\bar{K}\times ...\times\bar{K}</math>.
+Let {{mvar|K}} be a [[field (mathematics)|field]].  Let {{mvar|L}} be a [[commutative]] [[unital algebra|unital]] [[associative]] {{mvar|K}}-algebra.  Then {{mvar|L}} is called an ''étale {{mvar|K}}-algebra'' if any one of the following equivalent conditions holds:<ref>{{harv|Bourbaki|1990|loc=page A.V.28-30}}</ref>
+{{bulleted list
+|<math>L\otimes_{K} E\simeq E^n</math> for some field extension {{mvar|E}} of {{mvar|K}} and some nonnegative integer {{mvar|n}}.
+|<math>L\otimes_{K} \overline{K} \simeq \overline{K}^n</math> for any algebraic closure <math>\overline{K}</math> of {{mvar|K}} and some nonnegative integer {{mvar|n}}.
+|{{mvar|L}} is isomorphic to a finite product of finite separable field extensions of {{mvar|K}}.
+|{{mvar|L}} is finite-dimensional over {{mvar|K}}, and the trace form {{math|Tr(''xy'')}} is nondegenerate.
+|The morphism of schemes <math>\operatorname{Spec} L \to \operatorname{Spec} K</math> is an [[étale morphism]].
+}}
+==Examples==
-The name "étale algebra" comes from the fact that a finite dimensional commutative algebra over a field algebra is étale if and only if  <math>\mathrm{Spec}\,\mathfrak{R} \to \mathrm{Spec}\,K</math> is an [[étale morphism]].
+The <math>\mathbb{Q}</math>-algebra <math>\mathbb{Q}(i)</math> is étale because it is a finite separable field extension.
+The <math>\mathbb{R}</math>-algebra <math>\mathbb{R}[x]/(x^2)</math> is not étale, since <math>\mathbb{R}[x]/(x^2)\otimes_\mathbb{R}\mathbb{C} \simeq \mathbb{C}[x]/(x^2)</math>.
+==Properties==
+Let {{mvar|G}} denote the [[absolute Galois group]] of {{mvar|K}}.  Then the category of étale {{mvar|K}}-algebras is equivalent to the category of finite {{mvar|G}}-sets with continuous {{mvar|G}}-action.  In particular, étale algebras of dimension {{mvar|n}} are classified by [[conjugacy class]]es of continuous [[homomorphism]]s from {{mvar|G}} to the symmetric group {{math|''S''<sub>''n''</sub>}}. These globalize to e.g. the definition of [[Étale fundamental group|étale fundamental groups]] and categorify to [[Grothendieck's Galois theory]].
+== Notes ==
+{{Reflist}}
 ==References==
-*{{citation|MR=1080964
+*{{citation|mr=1080964
-|first=Bourbaki|last= N.
+|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
 {{DEFAULTSORT:Etale Algebra}}
-[[Category:Algebra]]
+[[Category:Commutative algebra]]