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Examples

• The polynomial algebra K [ x 1 , … , x n ] {\displaystyle K[x_{1},\dots ,x_{n}]} is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated.
• The field E = K ( t ) {\displaystyle E=K(t)} of rational functions in one variable over an infinite field K {\displaystyle K} is not a finitely generated algebra over K {\displaystyle K} . On the other hand, E {\displaystyle E} is generated over K {\displaystyle K} by a single element, t {\displaystyle t} , as a field.
• If E / F {\displaystyle E/F} is a finite field extension then it follows from the definitions that E {\displaystyle E} is a finitely generated algebra over F {\displaystyle F} .
• Conversely, if E / F {\displaystyle E/F} is a field extension and E {\displaystyle E} is a finitely generated algebra over F {\displaystyle F} then the field extension is finite. This is called Zariski's lemma. See also integral extension.
• If G {\displaystyle G} is a finitely generated group then the group algebra K G {\displaystyle KG} is a finitely generated algebra over K {\displaystyle 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 V ⊆ A n {\displaystyle V\subseteq \mathbb {A} ^{n}} we can associate a finitely generated K {\displaystyle K} -algebra

Γ ( V ) := K [ X 1 , … , X n ] / I ( V ) {\displaystyle \Gamma (V):=K[X_{1},\dots ,X_{n}]/I(V)}

called the affine coordinate ring of V {\displaystyle V} ; moreover, if ϕ : V → W {\displaystyle \phi \colon V\to W} is a regular map between the affine algebraic sets V ⊆ A n {\displaystyle V\subseteq \mathbb {A} ^{n}} and W ⊆ A m {\displaystyle W\subseteq \mathbb {A} ^{m}} , we can define a homomorphism of K {\displaystyle K} -algebras

Γ ( ϕ ) ≡ ϕ ∗ : Γ ( W ) → Γ ( V ) , ϕ ∗ ( f ) = f ∘ ϕ , {\displaystyle \Gamma (\phi )\equiv \phi ^{*}\colon \Gamma (W)\to \Gamma (V),\,\phi ^{*}(f)=f\circ \phi ,}

then, Γ {\displaystyle \Gamma } is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated K {\displaystyle K} -algebras: this functor turns out² to be an equivalence of categories

Γ : ( affine algebraic sets ) o p p → ( reduced finitely generated  K -algebras ) , {\displaystyle \Gamma \colon ({\text{affine algebraic sets}})^{\rm {opp}}\to ({\text{reduced finitely generated }}K{\text{-algebras}}),}

and, restricting to affine varieties (i.e. irreducible affine algebraic sets),

Γ : ( affine algebraic varieties ) o p p → ( integral finitely generated  K -algebras ) . {\displaystyle \Gamma \colon ({\text{affine algebraic varieties}})^{\rm {opp}}\to ({\text{integral finitely generated }}K{\text{-algebras}}).}

Finite algebras vs algebras of finite type

We recall that a commutative R {\displaystyle R} -algebra A {\displaystyle A} is a ring homomorphism ϕ : R → A {\displaystyle \phi \colon R\to A} ; the R {\displaystyle R} -module structure of A {\displaystyle A} is defined by

λ ⋅ a := ϕ ( λ ) a , λ ∈ R , a ∈ A . {\displaystyle \lambda \cdot a:=\phi (\lambda )a,\quad \lambda \in R,a\in A.}

An R {\displaystyle R} -algebra A {\displaystyle A} is called finite if it is finitely generated as an R {\displaystyle R} -module, i.e. there is a surjective homomorphism of R {\displaystyle R} -modules

R ⊕ n ↠ A . {\displaystyle R^{\oplus _{n}}\twoheadrightarrow A.}

Again, there is a characterisation of finite algebras in terms of quotients³

An R {\displaystyle R} -algebra A {\displaystyle A} is finite if and only if it is isomorphic to a quotient R ⊕ n / M {\displaystyle R^{\oplus _{n}}/M} by an R {\displaystyle R} -submodule M ⊆ R {\displaystyle M\subseteq R} .

By definition, a finite R {\displaystyle R} -algebra is of finite type, but the converse is false: the polynomial ring R [ X ] {\displaystyle R[X]} is of finite type but not finite. However, if an R {\displaystyle R} -algebra is of finite type and integral, then it is finite. More precisely, A {\displaystyle A} is a finitely generated R {\displaystyle R} -module if and only if A {\displaystyle A} is generated as an R {\displaystyle R} -algebra by a finite number of elements integral over R {\displaystyle R} .

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

See also

• Finitely generated module
• Finitely generated field extension
• Artin–Tate lemma
• Finite algebra
• Morphism of finite type

References

1. Kemper, Gregor (2009). A Course in Commutative Algebra. Springer. p. 8. ISBN 978-3-642-03545-6. 978-3-642-03545-6 ↩

2. 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. 978-3-8348-0676-5 ↩

3. Atiyah, Michael Francis; Macdonald, Ian Grant (1994). Introduction to commutative algebra. CRC Press. p. 21. ISBN 9780201407518. 9780201407518 ↩