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 $a_1,\dots ,a_n\in A$ s.t. the evaluation homomorphism at $\mathbf{𝐚}=(a_1,\dots ,a_n)$

${\varphi }_{\mathbf{𝐚}}:K[X_1,\dots ,X_n]\twoheadrightarrow A$

is surjective; thus, by applying the first isomorphism theorem, $A\simeq K[X_1,\dots ,X_n]/\mathrm{k}\mathrm{e}\mathrm{r}({\varphi }_{\mathbf{𝐚}})$.

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

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

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 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 $V\subset {\mathbb{𝔸}}^n$ we can associate a finitely generated $K$-algebra

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

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

$\mathrm{\Gamma }(\varphi )\equiv {\varphi }^{*}:\mathrm{\Gamma }(W)\to \mathrm{\Gamma }(V),{\varphi }^{*}(f)=f\circ \varphi ,$

then, $\mathrm{\Gamma }$ 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 to be an equivalence of categories

$\mathrm{\Gamma }:(\text{affine algebraic sets}{)}^{\mathrm{o}\mathrm{p}\mathrm{p}}\to (\text{reduced finitely generated }K\text{-algebras}),$

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

$\mathrm{\Gamma }:(\text{affine algebraic varieties}{)}^{\mathrm{o}\mathrm{p}\mathrm{p}}\to (\text{integral finitely generated }K\text{-algebras}).$

Finite algebras vs algebras of finite type

We recall that a commutative $R$-algebra $A$ is a ring homomorphism $\varphi :R\to A$; the $R$-module structure of $A$ is defined by

$\lambda \cdot a:=\varphi (\lambda )a,\lambda \in R,a\in A.$

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

$R^{{\oplus }_n}\twoheadrightarrow A.$

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

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

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