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,...,an of A such that every element of A can be expressed as a polynomial in a1,...,an, with coefficients in K.

Equivalently, there exist elements $$a_1,\dots,a_n\in A$$ such that the evaluation homomorphism at $${\bf a}=(a_1,\dots,a_n)$$
 * $$\phi_{\bf a}\colon K[X_1,\dots,X_n]\twoheadrightarrow A$$

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

Conversely, $$A:= K[X_1,\dots,X_n]/I$$ for any ideal $$ I\subseteq K[X_1,\dots,X_n]$$ 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 $$I\subseteq K[X_1,\dots,X_n]$$.

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 K[x1,...,xn&hairsp;] 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 $$V\subseteq \mathbb{A}^n$$ we can associate a finitely generated $$K$$-algebra
 * $$\Gamma(V):=K[X_1,\dots,X_n]/I(V)$$

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

then, $$\Gamma$$ 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
 * $$\Gamma\colon

(\text{affine algebraic sets})^{\rm opp}\to(\text{reduced finitely generated }K\text{-algebras}),$$ and, restricting to affine varieties (i.e. irreducible affine algebraic sets),
 * $$\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 $$R$$-algebra $$A$$ is a ring homomorphism $$\phi\colon R\to A$$; the $$R$$-module structure of $$A$$ is defined by
 * $$ \lambda \cdot a := \phi(\lambda)a,\quad\lambda\in R, a\in A.$$

An $$R$$-algebra $$A$$ is called 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 $$M\subseteq R$$.

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.