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.

In the definition above, one may replace the coefficient field K by an arbitrary commutative ring R, to arrive at the concept of an algebra finitely generated (or of finite type) over R.

• 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 ${\displaystyle K}$.

• 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.

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),

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

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.

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