In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A over a field K where there exists a finite set of elements a1,...,an of A such that every element of A can be expressed as a polynomial in a1,...,an, with coefficients in K.
Equivalently, there exist elements such that the evaluation homomorphism at
is surjective; thus, by applying the first isomorphism theorem, .
Conversely, for any ideal is a -algebra of finite type, indeed any element of is a polynomial in the cosets with coefficients in . Therefore, we obtain the following characterisation of finitely generated -algebras[1]
- is a finitely generated -algebra if and only if it is isomorphic as a -algebra to a quotient ring of the type by an ideal .
If it is necessary to emphasize the field K then the algebra is said to be finitely generated over K. Algebras that are not finitely generated are called infinitely generated.
Relation with affine varieties
[edit]Finitely generated reduced commutative algebras are basic objects of consideration in modern algebraic geometry, where they correspond to affine algebraic varieties; for this reason, these algebras are also referred to as (commutative) affine algebras. More precisely, given an affine algebraic set we can associate a finitely generated -algebra
called the affine coordinate ring of ; moreover, if is a regular map between the affine algebraic sets and , we can define a homomorphism of -algebras
then, is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated -algebras: this functor turns out[2] to be an equivalence of categories
and, restricting to affine varieties (i.e. irreducible affine algebraic sets),
Finite algebras vs algebras of finite type
[edit]We recall that a commutative -algebra is a ring homomorphism ; the -module structure of is defined by
An -algebra is called finite if it is finitely generated as an -module, i.e. there is a surjective homomorphism of -modules
Again, there is a characterisation of finite algebras in terms of quotients[3]
- An -algebra is finite if and only if it is isomorphic to a quotient by an -submodule .
By definition, a finite -algebra is of finite type, but the converse is false: the polynomial ring is of finite type but not finite.
Finite algebras and algebras of finite type are related to the notions of finite morphisms and morphisms of finite type.