In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
, an étale algebra over a
field is a special type of
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, one that is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to a finite product of
finite separable field extensions. An étale algebra is a special sort of commutative
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 homomorphism of (unital, but not necessarily ...
.
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. Perhaps most familiar as a pr ...
unital associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
-algebra. Then is called an ''étale -algebra'' if any one of the following equivalent conditions holds:
Examples
The
-algebra
is étale because it is a finite separable field extension.
The
-algebra
of
dual number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
D ...
s is not étale, since
.
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
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* 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 ...
es of continuous
group homomorphism
In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
whe ...
s from to the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
. These globalize to e.g. the definition of
étale fundamental groups and categorify to
Grothendieck's Galois theory In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in th ...
.
Notes
References
*
* http://www.jmilne.org/math/CourseNotes/FT.pdf
{{DEFAULTSORT:Etale Algebra
Commutative algebra