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
- 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.
- The polynomial algebra K[x1,...,xn] is finitely generated. The polynomial algebra in infinitely countably 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 ring KG is a finitely generated algebra over K.
- 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 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 outto be an equivalence of categories
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
- 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.
- 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. ISBN 978-3-8348-0676-5.
- Atiyah, Michael Francis; MacDonald, Ian Grant (1994). Introduction to commutative algebra. CRC Press. p. 21. ISBN 9780201407518.
- Finitely generated module
- Finitely generated field extension
- Artin–Tate lemma
- Finite algebra
- Morphisms of finite type