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 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 .
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 polynomial algebra K[x1,...,xn ] is finitely generated. The polynomial algebra in countably infinitely 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 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 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 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
- ^ 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. doi:10.1007/978-3-8348-9722-0. ISBN 978-3-8348-0676-5.
- ^ Atiyah, Michael Francis; Macdonald, Ian Grant (1994). Introduction to commutative algebra. CRC Press. p. 21. ISBN 9780201407518.