Finitely generated algebra

{{Short description|Type of algebra}} In mathematics, a '''finitely generated algebra''' (also called an '''algebra of finite type''') over a (commutative) ring <math>R</math>, or a finitely generated <math>R</math>-algebra for short, is a commutative associative algebra ''<math>A</math>'' defined by ring homomorphism <math>f:R\to A</math>, such that every element of ''<math>A</math>'' can be expressed as a polynomial in a finite number of generators <math>a_1,\dots,a_n\in A</math> with coefficients in <math>f(R)</math>. Put another way, there is a surjective <math>R</math>-algebra homomorphism from the polynomial ring <math>R[X_1,\dots,X_n]</math> to <math>A</math>.

If <math>K</math> is a field, regarded as a subalgebra of <math>A</math>, and <math>f</math> is the natural injection <math>K\hookrightarrow A</math>, then a <math>K</math>-algebra of finite type is a commutative associative algebra <math>A</math> where there exists a finite set of elements <math>a_1,\dots,a_n\in A</math> such that every element of ''<math>A</math>'' can be expressed as a polynomial in <math>a_1,\dots,a_n</math>, with coefficients in ''<math>K</math>''.

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 \cong 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:<ref>{{cite book |last=Kemper |first=Gregor |date=2009 |title=A Course in Commutative Algebra |url= https://www.springer.com/gp/book/9783642035449|publisher=Springer |page= 8|isbn=978-3-642-03545-6 }}</ref> :<math>A</math> is a finitely generated <math>K</math>-algebra if and only if it is isomorphic as a <math>K</math>-algebra 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>

Algebras that are not finitely generated are called '''infinitely generated'''.

A '''finitely generated ring''' refers to a ring that is finitely generated when it is regarded as a <math>\mathbb{Z}</math>-algebra.

An algebra being ''finitely generated'' (''of finite type'') should not be confused with an algebra being ''finite'' (see below). A '''finite algebra''' over <math>R</math> is a commutative associative algebra <math>A</math> that is ''finitely generated as a module''; that is, an <math>R</math>-algebra defined by ring homomorphism <math>f:R\to A</math>, such that every element of <math>A</math> can be expressed as a ''linear combination'' of a finite number of generators <math display="inline">a_1,\dots,a_n \in A</math> with coefficients in <math>f(R)</math>. This is a stronger condition than <math>A</math> being expressible as a ''polynomial'' in a finite set of generators in the case of the algebra being finitely generated.

== Examples ==

* The polynomial algebra <math>K[x_1,\dots,x_n]</math> is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated. * The ring of real-coefficient polynomials <math>\Bbb{R}[x]</math> is finitely generated over <math>\Bbb{R}</math> but not over <math>\Bbb{Q}</math>. * The field <math>E=K(t)</math> of rational functions in one variable over an infinite field ''<math>K</math>'' is ''not'' a finitely generated algebra over ''<math>K</math>''. On the other hand, <math>E</math> is generated over <math>K</math> by a single element, ''<math>t</math>'', ''as a field''. * If <math>E/F</math> is a finite field extension then it follows from the definitions that <math>E</math> is a finitely generated algebra over <math>F</math>. * Conversely, if <math>E/F</math> is a field extension and <math>E</math> is a finitely generated algebra over <math>F</math> then the field extension is finite. This is called Zariski's lemma. See also integral extension. * If <math>G</math> is a finitely generated group then the group algebra <math>KG</math> is a finitely generated algebra over <math>K</math>.

== 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 <math>A</math> is a finitely generated commutative algebra over a Noetherian ring then every ideal of ''A'' is finitely generated, or equivalently, ''<math>A</math>'' 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<ref>{{cite book |last1=Görtz |author-link1=Ulrich Görtz |last2=Wedhorn |first1=Ulrich |first2=Torsten |date=2010 |title=Algebraic Geometry I. Schemes With Examples and Exercises |url= https://link.springer.com/book/10.1007/978-3-8348-9722-0|publisher=Springer |page= 19|doi=10.1007/978-3-8348-9722-0 |isbn=978-3-8348-0676-5}}</ref> 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:<ref>{{cite book |last1=Atiyah|last2=Macdonald |first1=Michael Francis|first2=Ian Grant|author1link = Michael Atiyah|author2link = Ian G. Macdonald |date=1994 |title=Introduction to commutative algebra |url=https://www.crcpress.com/Introduction-To-Commutative-Algebra/Atiyah/p/book/9780201407518 |publisher=CRC Press |page= 21|isbn=9780201407518}}</ref> :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. However, if an <math>R</math>-algebra is of finite type and integral, then it is finite. More precisely, <math>A</math> is a finitely generated <math>R</math>-module if and only if <math>A</math> is generated as an <math>R</math>-algebra by a finite number of elements integral over <math>R</math>.

Finite algebras and algebras of finite type are related to the notions of finite morphisms and morphisms of finite type.

== References == {{Reflist}}

Category:Algebras Category:Commutative algebra