Finitely generated algebra

In mathematics, a finitely generated algebra (also called an algebra of finite type) over a (commutative) ring ${\displaystyle R}$ , or a finitely generated ${\displaystyle R}$ -algebra for short, is a commutative associative algebra ${\displaystyle A}$ defined by ring homomorphism ${\displaystyle f:R\to A}$ , such that every element of ${\displaystyle A}$ can be expressed as a polynomial in a finite number of generators ${\displaystyle a_{1},\dots ,a_{n}\in A}$ with coefficients in ${\displaystyle f(R)}$ . Put another way, there is a surjective ${\displaystyle R}$ -algebra homomorphism from the polynomial ring ${\displaystyle R[X_{1},\dots ,X_{n}]}$ to ${\displaystyle A}$ .

If ${\displaystyle K}$ is a field, regarded as a subalgebra of ${\displaystyle A}$ , and ${\displaystyle f}$ is the natural injection ${\displaystyle K\hookrightarrow A}$ , then a ${\displaystyle K}$ -algebra of finite type is a commutative associative algebra ${\displaystyle A}$ where there exists a finite set of elements ${\displaystyle a_{1},\dots ,a_{n}\in A}$ such that every element of ${\displaystyle A}$ can be expressed as a polynomial in ${\displaystyle a_{1},\dots ,a_{n}}$ , with coefficients in ${\displaystyle K}$ .

Equivalently, there exist elements ${\displaystyle a_{1},\dots ,a_{n}\in A}$ such that the evaluation homomorphism at ${\displaystyle {\bf {a}}=(a_{1},\dots ,a_{n})}$

${\displaystyle \phi _{\bf {a}}\colon K[X_{1},\dots ,X_{n}]\twoheadrightarrow A}$

is surjective; thus, by applying the first isomorphism theorem, ${\displaystyle A\cong K[X_{1},\dots ,X_{n}]/{\rm {ker}}(\phi _{\bf {a}})}$ .

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

${\displaystyle A}$ is a finitely generated ${\displaystyle K}$ -algebra if and only if it is isomorphic as a ${\displaystyle K}$ -algebra to a quotient ring of the type ${\displaystyle K[X_{1},\dots ,X_{n}]/I}$ by an ideal ${\displaystyle I\subseteq K[X_{1},\dots ,X_{n}].}$

Algebras that are not finitely generated are called infinitely generated.

A finitely generated ring refers to a ring that is finitely generated when it is regarded as a ${\displaystyle \mathbb {Z} }$ -algebra.

An algebra being finitely generated (of finite type) should not be confused with an algebra being finite (see below). A finite algebra over ${\displaystyle R}$ is a commutative associative algebra ${\displaystyle A}$ that is finitely generated as a module; that is, an ${\displaystyle R}$ -algebra defined by ring homomorphism ${\displaystyle f:R\to A}$ , such that every element of ${\displaystyle A}$ can be expressed as a linear combination of a finite number of generators ${\textstyle a_{1},\dots ,a_{n}\in A}$ with coefficients in ${\displaystyle f(R)}$ . This is a stronger condition than ${\displaystyle A}$ being expressible as a polynomial in a finite set of generators in the case of the algebra being finitely generated.

Examples

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

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 ${\displaystyle A}$ is a finitely generated commutative algebra over a Noetherian ring then every ideal of A is finitely generated, or equivalently, ${\displaystyle A}$ 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 ${\displaystyle V\subseteq \mathbb {A} ^{n}}$ we can associate a finitely generated ${\displaystyle K}$ -algebra

${\displaystyle \Gamma (V):=K[X_{1},\dots ,X_{n}]/I(V)}$

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

${\displaystyle \Gamma (\phi )\equiv \phi ^{*}\colon \Gamma (W)\to \Gamma (V),\,\phi ^{*}(f)=f\circ \phi ,}$

then, ${\displaystyle \Gamma }$ is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated ${\displaystyle K}$ -algebras: this functor turns out^[2] to be an equivalence of categories

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

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

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

Finite algebras vs algebras of finite type

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

${\displaystyle \lambda \cdot a:=\phi (\lambda )a,\quad \lambda \in R,a\in A.}$

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

${\displaystyle R^{\oplus _{n}}\twoheadrightarrow A.}$

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

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

By definition, a finite ${\displaystyle R}$ -algebra is of finite type, but the converse is false: the polynomial ring ${\displaystyle R[X]}$ is of finite type but not finite. However, if an ${\displaystyle R}$ -algebra is of finite type and integral, then it is finite. More precisely, ${\displaystyle A}$ is a finitely generated ${\displaystyle R}$ -module if and only if ${\displaystyle A}$ is generated as an ${\displaystyle R}$ -algebra by a finite number of elements integral over ${\displaystyle R}$ .

Finite algebras and algebras of finite type are related to the notions of finite morphisms and morphisms of finite type.

References

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

See also

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