Английская Википедия: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 <math>a_1,\dots,a_n\in A</math> such that the evaluation homomorphism at <math>{\bf a}=(a_1,\dots,a_n)</math>
- <math>\phi_{\bf a}\colon K[X_1,\dots,X_n]\twoheadrightarrow A</math>
is surjective; thus, by applying the first isomorphism theorem, <math>A \simeq K[X_1,\dots,X_n]/{\rm ker}(\phi_{\bf a})</math>.
Conversely, <math>A:= K[X_1,\dots,X_n]/I</math> for any ideal <math> I\subseteq K[X_1,\dots,X_n]</math> is a <math>K</math>-algebra of finite type, indeed any element of <math>A</math> is a polynomial in the cosets <math>a_i:=X_i+I, i=1,\dots,n</math> with coefficients in <math>K</math>. Therefore, we obtain the following characterisation of finitely generated <math>K</math>-algebras[1]
- <math>A</math> is a finitely generated <math>K</math>-algebra if and only if it is isomorphic to a quotient ring of the type <math>K[X_1,\dots,X_n]/I</math> by an ideal <math>I\subseteq K[X_1,\dots,X_n]</math>.
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 ] 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 <math>V\subseteq \mathbb{A}^n</math> we can associate a finitely generated <math>K</math>-algebra
- <math>\Gamma(V):=K[X_1,\dots,X_n]/I(V)</math>
called the affine coordinate ring of <math>V</math>; moreover, if <math>\phi\colon V\to W</math> is a regular map between the affine algebraic sets <math>V\subseteq \mathbb{A}^n</math> and <math>W\subseteq \mathbb{A}^m</math>, we can define a homomorphism of <math>K</math>-algebras
- <math>\Gamma(\phi)\equiv\phi^*\colon\Gamma(W)\to\Gamma(V),\,\phi^*(f)=f\circ\phi,</math>
then, <math>\Gamma</math> is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated <math>K</math>-algebras: this functor turns out[2] to be an equivalence of categories
- <math>\Gamma\colon
(\text{affine algebraic sets})^{\rm opp}\to(\text{reduced finitely generated }K\text{-algebras}),</math> and, restricting to affine varieties (i.e. irreducible affine algebraic sets),
- <math>\Gamma\colon
(\text{affine algebraic varieties})^{\rm opp}\to(\text{integral finitely generated }K\text{-algebras}).</math>
Finite algebras vs algebras of finite type
We recall that a commutative <math>R</math>-algebra <math>A</math> is a ring homomorphism <math>\phi\colon R\to A</math>; the <math>R</math>-module structure of <math>A</math> is defined by
- <math> \lambda \cdot a := \phi(\lambda)a,\quad\lambda\in R, a\in A.</math>
An <math>R</math>-algebra <math>A</math> is called finite if it is finitely generated as an <math>R</math>-module, i.e. there is a surjective homomorphism of <math>R</math>-modules
- <math> R^{\oplus_n}\twoheadrightarrow A.</math>
Again, there is a characterisation of finite algebras in terms of quotients[3]
- An <math>R</math>-algebra <math>A</math> is finite if and only if it is isomorphic to a quotient <math>R^{\oplus_n}/M</math> by an <math>R</math>-submodule <math>M\subseteq R</math>.
By definition, a finite <math>R</math>-algebra is of finite type, but the converse is false: the polynomial ring <math>R[X]</math> 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.
References
See also
- Finitely generated module
- Finitely generated field extension
- Artin–Tate lemma
- Finite algebra
- Morphism of finite type
