Finitely generated algebra
In mathematics, a finitely generated algebra over a ring, or a finitely generated -algebra for short, is a commutative associative algebra ' defined by ring homomorphism, such that every element 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:
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. A finite algebra over is a commutative associative algebra that is finitely generated as a module; that is, an -algebra defined by ring homomorphism, such that every element of can be expressed as a linear combination of a finite number of generators with coefficients in. This is a stronger condition than being expressible as a polynomial in a finite set of generators in the case of the algebra being finitely generated.
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.
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 is a finitely generated commutative algebra over a Noetherian ring then every ideal of A is finitely generated, or equivalently, is a Noetherian ring.
Relation with affine varieties
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 to be an equivalence of categories
and, restricting to affine varieties,
Finite algebras vs algebras of finite type
We recall that a commutative -algebra is a ring homomorphism ; the -module structure of is defined byAn -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:
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.