Finitely generated algebra

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 [math]\displaystyle{ a_1,\dots,a_n\in A }[/math] s.t. the evaluation homomorphism at [math]\displaystyle{ {\bf a}=(a_1,\dots,a_n) }[/math]

[math]\displaystyle{ \phi_{\bf a}\colon K[X_1,\dots,X_n]\twoheadrightarrow A }[/math]

is surjective; thus, by applying the first isomorphism theorem, [math]\displaystyle{ A \simeq K[X_1,\dots,X_n]/{\rm ker}(\phi_{\bf a}) }[/math].

Conversely, [math]\displaystyle{ A:= K[X_1,\dots,X_n]/I }[/math] for any ideal [math]\displaystyle{ I\subset K[X_1,\dots,X_n] }[/math] is a [math]\displaystyle{ K }[/math]-algebra of finite type, indeed any element of [math]\displaystyle{ A }[/math] is a polynomial in the cosets [math]\displaystyle{ a_i:=X_i+I, i=1,\dots,n }[/math] with coefficients in [math]\displaystyle{ K }[/math]. Therefore, we obtain the following characterisation of finitely generated [math]\displaystyle{ K }[/math]-algebras

[math]\displaystyle{ A }[/math] is a finitely generated [math]\displaystyle{ K }[/math]-algebra if and only if it is isomorphic to a quotient ring of the type [math]\displaystyle{ K[X_1,\dots,X_n]/I }[/math] by an ideal [math]\displaystyle{ I\subset K[X_1,\dots,X_n] }[/math].

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[x₁,...,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 [math]\displaystyle{ V\subset \mathbb{A}^n }[/math] we can associate a finitely generated [math]\displaystyle{ K }[/math]-algebra

[math]\displaystyle{ \Gamma(V):=K[X_1,\dots,X_n]/I(V) }[/math]

called the affine coordinate ring of [math]\displaystyle{ V }[/math]; moreover, if [math]\displaystyle{ \phi\colon V\to W }[/math] is a regular map between the affine algebraic sets [math]\displaystyle{ V\subset \mathbb{A}^n }[/math] and [math]\displaystyle{ W\subset \mathbb{A}^m }[/math], we can define a homomorphism of [math]\displaystyle{ K }[/math]-algebras

[math]\displaystyle{ \Gamma(\phi)\equiv\phi^*\colon\Gamma(W)\to\Gamma(V),\,\phi^*(f)=f\circ\phi, }[/math]

then, [math]\displaystyle{ \Gamma }[/math] is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated [math]\displaystyle{ K }[/math]-algebras: this functor turns out to be an equivalence of categories

[math]\displaystyle{ \Gamma\colon (\text{affine algebraic sets})^{\rm opp}\to(\text{reduced finitely generated }K\text{-algebras}), }[/math]

and, restricting to affine varieties (i.e. irreducible affine algebraic sets),

[math]\displaystyle{ \Gamma\colon (\text{affine algebraic varieties})^{\rm opp}\to(\text{integral finitely generated }K\text{-algebras}). }[/math]

Finite algebras vs algebras of finite type

We recall that a commutative [math]\displaystyle{ R }[/math]-algebra [math]\displaystyle{ A }[/math] is a ring homomorphism [math]\displaystyle{ \phi\colon R\to A }[/math]; the [math]\displaystyle{ R }[/math]-module structure of [math]\displaystyle{ A }[/math] is defined by

[math]\displaystyle{ \lambda \cdot a := \phi(\lambda)a,\quad\lambda\in R, a\in A. }[/math]

An [math]\displaystyle{ R }[/math]-algebra [math]\displaystyle{ A }[/math] is finite if it is finitely generated as an [math]\displaystyle{ R }[/math]-module, i.e. there is a surjective homomorphism of [math]\displaystyle{ R }[/math]-modules

[math]\displaystyle{ R^{\oplus_n}\twoheadrightarrow A. }[/math]

Again, there is a characterisation of finite algebras in terms of quotients

An [math]\displaystyle{ R }[/math]-algebra [math]\displaystyle{ A }[/math] is finite if and only if it is isomorphic to a quotient [math]\displaystyle{ R^{\oplus_n}/M }[/math] by an [math]\displaystyle{ R }[/math]-submodule [math]\displaystyle{ M\subset R }[/math].

By definition, a finite [math]\displaystyle{ R }[/math]-algebra is of finite type, but the converse is false: the polynomial ring [math]\displaystyle{ R[X] }[/math] 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