Finitely generated algebra
In
Equivalently, there exist elements such that the evaluation homomorphism at
is
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 ringof 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
- The countably infinitely manygenerators 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 extensionthen 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 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
called the affine
then, is a
and, restricting to
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 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.
Finite algebras and algebras of finite type are related to the notions of finite morphisms and morphisms of finite type.
References
- ISBN 978-3-642-03545-6.
- ISBN 978-3-8348-0676-5.
- ISBN 9780201407518.
See also
- Finitely generated module
- Finitely generated field extension
- Artin–Tate lemma
- Finite algebra
- Morphism of finite type