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Finitely generated algebra

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In mathematics, a finitely generated algebra (also called an algebra of finite type) over a (commutative) ring , or a finitely generated -algebra for short, is a commutative associative algebra where, given a ring homomorphism , all elements of can be expressed as a polynomial in a finite number of generators with coefficients in . Put another way, there is a surjective -algebra homomorphism from the polynomial ring to .

If is a field, regarded as a subalgebra of , and is the natural injection , then a -algebra of finite type is a commutative associative algebra where there exists a finite set of elements such that every element of can be expressed as a polynomial in , with coefficients in .

Equivalently, there exist elements such that the evaluation homomorphism at

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

Conversely, for any ideal is a -algebra of finite type, indeed any element of is a polynomial in the cosets with coefficients in . Therefore, we obtain the following characterisation of finitely generated -algebras[1]

is a finitely generated -algebra if and only if it is isomorphic as a -algebra to a quotient ring of the type by an ideal .

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 -algebra.

An algebra being finitely generated should not be confused with an algebra being finite (see below). A finite algebra over is a commutative associative algebra that is finitely generated as a module; that is, there exists a finite set of generators such that every element of can be expressed as a linear combination of with the coefficients in .

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Examples

  • The polynomial algebra is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated.
  • The ring of real-coefficient polynomials is finitely generated over but not over .
  • The field of rational functions in one variable over an infinite field is not a finitely generated algebra over . On the other hand, is generated over by a single element, , as a field.
  • If is a finite field extension then it follows from the definitions that is a finitely generated algebra over .
  • Conversely, if is a field extension and is a finitely generated algebra over then the field extension is finite. This is called Zariski's lemma. See also integral extension.
  • If is a finitely generated group then the group algebra is a finitely generated algebra over .
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Properties

Relation with affine varieties

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Perspective

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 we can associate a finitely generated -algebra

called the affine coordinate ring of ; moreover, if is a regular map between the affine algebraic sets and , we can define a homomorphism of -algebras

then, is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated -algebras: this functor turns out[2] to be an equivalence of categories

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

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Finite algebras vs algebras of finite type

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Perspective

We recall that a commutative -algebra is a ring homomorphism ; the -module structure of is defined by

An -algebra is called finite if it is finitely generated as an -module, i.e. there is a surjective homomorphism of -modules

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

An -algebra is finite if and only if it is isomorphic to a quotient by an -submodule .

By definition, a finite -algebra is of finite type, but the converse is false: the polynomial ring is of finite type but not finite. However, if an -algebra is of finite type and integral, then it is finite. More precisely, is a finitely generated -module if and only if is generated as an -algebra by a finite number of elements integral over .

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

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References

See also

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