commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...

, an étale algebra over a field is a special type of

algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

, one that is isomorphic to a finite product of finite separable field extensions. These may also be called

separable algebra In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension. Definition and First Properties A ring homomorphism (of unital, but not necess ...

s, though the latter term is sometimes used in a broader sense.

Definitions

Let be a field. Let be a

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

unital

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

-algebra. Then is called an ''étale -algebra'' if any one of the following equivalent conditions holds:

Examples

The

\mathbb

-algebra

\mathbb(i)

is étale because it is a finite separable field extension. The

\mathbb

-algebra

\mathbb (x^2)

is not étale, since

\mathbb (x^2)\otimes_\mathbb\mathbb \simeq \mathbb (x^2)

Properties

Let denote the

absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...

of . Then the category of étale -algebras is equivalent to the category of finite -sets with continuous -action. In particular, étale algebras of dimension are classified by

conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other w ...

es of continuous

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

s from to the symmetric group .

Notes

References

* * http://www.jmilne.org/math/CourseNotes/FT.pdf {{DEFAULTSORT:Etale Algebra Commutative algebra