Finitely generated algebra

In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra ${\displaystyle A}$ over a field ${\displaystyle K}$ where there exists a finite set of elements ${\displaystyle a_{1},\dots ,a_{n}}$ of ${\displaystyle A}$ 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, ${\displaystyle A\simeq K[X_{1},\dots ,X_{n}]/{\rm {ker}}(\phi _{\bf {a}})}$.

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 ${\displaystyle I\subseteq K[X_{1},\dots ,X_{n}]}$.

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 ${\displaystyle K[x_{1},\dots ,x_{n}]}$ is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated.
• 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 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 ${\displaystyle V\subseteq \mathbb {A} ^{n}}$ 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