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 ''a''₁,...,''a''_''n'' of ''A'' such that every element of ''A'' can be expressed as a polynomial in ''a''₁,...,''a''_''n'', with coefficients in ''K''.

Equivalently, there exist elements $a_1,\dots,a_n\in A$ s.t. the evaluation homomorphism at $=(a_1,\dots,a_n)$
:\phi_\colon K _1,\dots,X_ntwoheadrightarrow A
is surjective; thus, by applying the first isomorphism theorem, A \simeq K _1,\dots,X_n(\phi_).

Conversely, A:= K _1,\dots,X_nI for any ideal I\subset K _1,\dots,X_n/math> 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 _1,\dots,X_nI by an ideal I\subset K _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'' 'x''₁,...,''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 $V\subset \mathbb^n$ we can associate a finitely generated $K$-algebra
:\Gamma(V):=K _1,\dots,X_nI(V)
called the affine coordinate ring of $V$; moreover, if $\phi\colon V\to W$ is a regular map between the affine algebraic sets $V\subset \mathbb^n$ and $W\subset \mathbb^m$, 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
:$\Gamma\colon (\text)^\to(\textK\text),$
and, restricting to affine varieties (i.e. irreducible affine algebraic sets),
:$\Gamma\colon (\text)^\to(\textK\text).$

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 ''finite'' if it is finitely generated as an $R$-module, i.e. there is a surjective homomorphism of $R$-modules
:$R^\twoheadrightarrow A.$

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 $R^/M$ 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 /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

Algebras
Commutative algebra

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