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 $a_{1},\dots ,a_{n}\in A$ 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^[1]

A

is a finitely generated

K

-algebra if and only if it is isomorphic as a

K

-algebra 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,...,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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\subseteq \mathbb{A}^n} we can associate a finitely generated $K$ -algebra

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(V):=K[X_1,\dots,X_n]/I(V)}

called the affine coordinate ring of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} ; moreover, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi\colon V\to W} is a regular map between the affine algebraic sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\subseteq \mathbb{A}^n} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W\subseteq \mathbb{A}^m} , we can define a homomorphism of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} -algebras

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(\phi)\equiv\phi^*\colon\Gamma(W)\to\Gamma(V),\,\phi^*(f)=f\circ\phi,}

then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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^[2] to be an equivalence of categories

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a ring homomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi\colon R\to A} ; the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -module structure of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \cdot a := \phi(\lambda)a,\quad\lambda\in R, a\in A.}

An Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is called finite if it is finitely generated as an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -module, i.e. there is a surjective homomorphism of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -modules

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^{\oplus_n}\twoheadrightarrow A.}

Again, there is a characterisation of finite algebras in terms of quotients^[3]

An Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is finite if and only if it is isomorphic to a quotient

R^{\oplus _{n}}/M

by an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -submodule Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\subseteq R} .

By definition, a finite Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -algebra is of finite type, but the converse is false: the polynomial ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

References

^ Kemper, Gregor (2009). A Course in Commutative Algebra. Springer. p. 8. ISBN 978-3-642-03545-6.
^ Görtz, Ulrich; Wedhorn, Torsten (2010). Algebraic Geometry I. Schemes With Examples and Exercises. Springer. p. 19. doi:10.1007/978-3-8348-9722-0. ISBN 978-3-8348-0676-5.
^ Atiyah, Michael Francis; Macdonald, Ian Grant (1994). Introduction to commutative algebra. CRC Press. p. 21. ISBN 9780201407518.

Examples

Properties

Relation with affine varieties

Finite algebras vs algebras of finite type

References

See also