Finitely generated algebra explained

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,...,a_n of A such that every element of A can be expressed as a polynomial in a₁,...,a_n, with coefficients in K.

Equivalently, there exist elements

a _{1,...,a} _{n\in} A such that the evaluation homomorphism at {\bf a} = (a _{1,...,a} _{n)} \phi _{\bf} \colon K[X _{1,...,X} _{n]\twoheadrightarrow} A is surjective; thus, by applying the first isomorphism theorem, A \simeq K[X _{1,...,X} _{n]/{\rm} ker}(\phi _{\bf} ) .

Conversely,

A: = K[X _{1,...,X} _{n]/I} for any ideal I\subseteq K[X _{1,...,X} _{n]} is a K -algebra of finite type, indeed any element of A is a polynomial in the cosets a _{i:=X} _{i+I,} i =1 , ... ,n with coefficients in K . Therefore, we obtain the following characterisation of finitely generated K -algebras^[1] A is a finitely generated K -algebra if and only if it is isomorphic as a K -algebra to a quotient ring of the type K[X _{1,...,X} _{n]/I} by an ideal I\subseteq K[X _{1,...,X} _{n]} .

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.

Examples

• The polynomial algebra K[''x''₁,...,''x''_''n'' ] is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated.
• The field E = K(t) of rational functions in one variable over an infinite field K is not a finitely generated algebra over K. On the other hand, E is generated over K by a single element, t, as a field.
• If E/F is a finite field extension then it follows from the definitions that E is a finitely generated algebra over F.
• Conversely, if E/F is a field extension and E is a finitely generated algebra over F then the field extension is finite. This is called Zariski's lemma. See also integral extension.
• If G is a finitely generated group then the group algebra KG is a finitely generated algebra over K.

Properties

• A homomorphic image of a finitely generated algebra is itself finitely generated. However, a similar property for subalgebras does not hold in general.
• Hilbert's basis theorem: if A is a finitely generated commutative algebra over a Noetherian ring then every ideal of A is finitely generated, or equivalently, A is a Noetherian ring.

Relation with affine varieties

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

V\subseteq A ^{n} we can associate a finitely generated K -algebra\Gamma(V): = K[X _{1,...,X} _{n]/I(V)} called the affine coordinate ring of V ; moreover, if \phi\colon V\to W is a regular map between the affine algebraic sets V\subseteq A ^{n} and W\subseteq A ^{m} , we can define a homomorphism of K -algebras\Gamma(\phi)\equiv\phi ^{*\colon\Gamma(W)\to\Gamma(V),\phi} ^{*(f)=f\circ\phi,} then, \Gamma is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated K -algebras: this functor turns out^[2] to be an equivalence of categories\Gamma\colon ( affinealgebraicsets ) ^{\rm} \to( reducedfinitelygenerated K -algebras ), and, restricting to affine varieties (i.e. irreducible affine algebraic sets),\Gamma\colon ( affinealgebraicvarieties ) ^{\rm} \to( integralfinitelygenerated K -algebras ).

Finite algebras vs algebras of finite type

We recall that a commutative

R -algebra A is a ring homomorphism \phi\colon R\to A ; the R -module structure of A is defined byλ ⋅ a : = \phi( λ )a, λ \in R, a\in A.

An

R -algebra A is called finite if it is finitely generated as an R -module, i.e. there is a surjective homomorphism of R -modulesR ^{⊕ _{n}} \twoheadrightarrow A.

Again, there is a characterisation of finite algebras in terms of quotients^[3]

An

R -algebra A is finite if and only if it is isomorphic to a quotient R ^{⊕ _{n}} /M by an R -submodule M\subseteq R .

By definition, a finite

R -algebra is of finite type, but the converse is false: the polynomial ring R[X] 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.

See also

• Finitely generated module
• Finitely generated field extension
• Artin–Tate lemma
• Finite algebra
• Morphism of finite type

Notes and References

1. Book: Kemper, Gregor . 2009 . A Course in Commutative Algebra . Springer . 8. 978-3-642-03545-6 .
2. Book: Görtz . Ulrich Görtz . Wedhorn . Ulrich . Torsten . 2010 . Algebraic Geometry I. Schemes With Examples and Exercises . Springer . 19. 10.1007/978-3-8348-9722-0 . 978-3-8348-0676-5.
3. Book: Atiyah. Macdonald . Michael Francis. Ian Grant. Michael Atiyah. Ian G. Macdonald . 1994 . Introduction to commutative algebra . CRC Press . 21. 9780201407518.