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Finitely generated algebra

From Wikipedia, the free encyclopedia

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 s.t. 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 to a quotient ring of the type by an ideal .

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 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),

Finite algebras vs algebras of finite type

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

An -algebra is 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.

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

References

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

See also


This page was last edited on 23 December 2020, at 02:11
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