Finitely generated algebra explained

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

a1,...,an\inA

such that the evaluation homomorphism at

{\bfa}=(a1,...,an)

\phi\bf\colonK[X1,...,Xn]\twoheadrightarrowA

is surjective; thus, by applying the first isomorphism theorem,

A\simeqK[X1,...,Xn]/{\rmker}(\phi\bf)

.

Conversely,

A:=K[X1,...,Xn]/I

for any ideal

I\subseteqK[X1,...,Xn]

is a

K

-algebra of finite type, indeed any element of

A

is a polynomial in the cosets

ai:=Xi+I,i=1,...,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[X1,...,Xn]/I

by an ideal

I\subseteqK[X1,...,Xn]

.

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

Properties

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\subseteqAn

we can associate a finitely generated

K

-algebra

\Gamma(V):=K[X1,...,Xn]/I(V)

called the affine coordinate ring of

V

; moreover, if

\phi\colonV\toW

is a regular map between the affine algebraic sets

V\subseteqAn

and

W\subseteqAm

, 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[2] to be an equivalence of categories

\Gamma\colon (affinealgebraicsets)\rm\to(reducedfinitelygeneratedK-algebras),

and, restricting to affine varieties (i.e. irreducible affine algebraic sets),

\Gamma\colon (affinealgebraicvarieties)\rm\to(integralfinitelygeneratedK-algebras).

Finite algebras vs algebras of finite type

We recall that a commutative

R

-algebra

A

is a ring homomorphism

\phi\colonR\toA

; the

R

-module structure of

A

is defined by

λa:=\phi(λ)a,   λ\inR,a\inA.

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
n
R

\twoheadrightarrowA.

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

An

R

-algebra

A

is finite if and only if it is isomorphic to a quotient
n
R

/M

by an

R

-submodule

M\subseteqR

.

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.

See also

Notes and References

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