Finitely generated algebra

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In mathematics, a finitely generated algebra (also called an algebra of finite type) over a (commutative) ring R {\displaystyle R} {\displaystyle R}, or a finitely generated R {\displaystyle R} {\displaystyle R}-algebra for short, is a commutative associative algebra A {\displaystyle A} {\displaystyle A} defined by ring homomorphism f : R → A {\displaystyle f:R\to A} {\displaystyle f:R\to A}, such that every element of A {\displaystyle A} {\displaystyle A} can be expressed as a polynomial in a finite number of generators a 1 , … , a n ∈ A {\displaystyle a_{1},\dots ,a_{n}\in A} {\displaystyle a_{1},\dots ,a_{n}\in A} with coefficients in f ( R ) {\displaystyle f(R)} {\displaystyle f(R)}. Put another way, there is a surjective R {\displaystyle R} {\displaystyle R}-algebra homomorphism from the polynomial ring R [ X 1 , … , X n ] {\displaystyle R[X_{1},\dots ,X_{n}]} {\displaystyle R[X_{1},\dots ,X_{n}]} to A {\displaystyle A} {\displaystyle A}.

If K {\displaystyle K} {\displaystyle K} is a field, regarded as a subalgebra of A {\displaystyle A} {\displaystyle A}, and f {\displaystyle f} {\displaystyle f} is the natural injection K ↪ A {\displaystyle K\hookrightarrow A} {\displaystyle K\hookrightarrow A}, then a K {\displaystyle K} {\displaystyle K}-algebra of finite type is a commutative associative algebra A {\displaystyle A} {\displaystyle A} where there exists a finite set of elements a 1 , … , a n ∈ A {\displaystyle a_{1},\dots ,a_{n}\in A} {\displaystyle a_{1},\dots ,a_{n}\in A} such that every element of A {\displaystyle A} {\displaystyle A} can be expressed as a polynomial in a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} {\displaystyle a_{1},\dots ,a_{n}}, with coefficients in K {\displaystyle K} {\displaystyle K}.

Equivalently, there exist elements a 1 , … , a n ∈ A {\displaystyle a_{1},\dots ,a_{n}\in A} {\displaystyle a_{1},\dots ,a_{n}\in A} such that the evaluation homomorphism at a = ( a 1 , … , a n ) {\displaystyle {\bf {a}}=(a_{1},\dots ,a_{n})} {\displaystyle {\bf {a}}=(a_{1},\dots ,a_{n})}

ϕ a : K [ X 1 , … , X n ] ↠ A {\displaystyle \phi _{\bf {a}}\colon K[X_{1},\dots ,X_{n}]\twoheadrightarrow A} {\displaystyle \phi _{\bf {a}}\colon K[X_{1},\dots ,X_{n}]\twoheadrightarrow A}

is surjective; thus, by applying the first isomorphism theorem, A ≅ K [ X 1 , … , X n ] / k e r ( ϕ a ) {\displaystyle A\cong K[X_{1},\dots ,X_{n}]/{\rm {ker}}(\phi _{\bf {a}})} {\displaystyle A\cong K[X_{1},\dots ,X_{n}]/{\rm {ker}}(\phi _{\bf {a}})}.

Conversely, A := K [ X 1 , … , X n ] / I {\displaystyle A:=K[X_{1},\dots ,X_{n}]/I} {\displaystyle A:=K[X_{1},\dots ,X_{n}]/I} for any ideal I ⊆ K [ X 1 , … , X n ] {\displaystyle I\subseteq K[X_{1},\dots ,X_{n}]} {\displaystyle I\subseteq K[X_{1},\dots ,X_{n}]} is a K {\displaystyle K} {\displaystyle K}-algebra of finite type, indeed any element of A {\displaystyle A} {\displaystyle A} is a polynomial in the cosets a i := X i + I , i = 1 , … , n {\displaystyle a_{i}:=X_{i}+I,i=1,\dots ,n} {\displaystyle a_{i}:=X_{i}+I,i=1,\dots ,n} with coefficients in K {\displaystyle K} {\displaystyle K}. Therefore, we obtain the following characterisation of finitely generated K {\displaystyle K} {\displaystyle K}-algebras:[1]

A {\displaystyle A} {\displaystyle A} is a finitely generated K {\displaystyle K} {\displaystyle K}-algebra if and only if it is isomorphic as a K {\displaystyle K} {\displaystyle K}-algebra to a quotient ring of the type K [ X 1 , … , X n ] / I {\displaystyle K[X_{1},\dots ,X_{n}]/I} {\displaystyle K[X_{1},\dots ,X_{n}]/I} by an ideal I ⊆ K [ X 1 , … , X n ] . {\displaystyle I\subseteq K[X_{1},\dots ,X_{n}].} {\displaystyle I\subseteq K[X_{1},\dots ,X_{n}].}

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 Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }-algebra.

An algebra being finitely generated (of finite type) should not be confused with an algebra being finite (see below). A finite algebra over R {\displaystyle R} {\displaystyle R} is a commutative associative algebra A {\displaystyle A} {\displaystyle A} that is finitely generated as a module; that is, an R {\displaystyle R} {\displaystyle R}-algebra defined by ring homomorphism f : R → A {\displaystyle f:R\to A} {\displaystyle f:R\to A}, such that every element of A {\displaystyle A} {\displaystyle A} can be expressed as a linear combination of a finite number of generators a 1 , … , a n ∈ A {\textstyle a_{1},\dots ,a_{n}\in A} {\textstyle a_{1},\dots ,a_{n}\in A} with coefficients in f ( R ) {\displaystyle f(R)} {\displaystyle f(R)}. This is a stronger condition than A {\displaystyle A} {\displaystyle A} being expressible as a polynomial in a finite set of generators in the case of the algebra being finitely generated.

Examples

  • The polynomial algebra K [ X 1 , … , X n ] {\displaystyle K[X_{1},\dots ,X_{n}]} {\displaystyle K[X_{1},\dots ,X_{n}]} is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated.
  • The ring of real-coefficient polynomials R [ X ] {\displaystyle {\mathbb {R}}[X]} {\displaystyle {\mathbb {R}}[X]} is finitely generated over R {\displaystyle {\mathbb {R}}} {\displaystyle {\mathbb {R}}} but not over Q {\displaystyle {\mathbb {Q}}} {\displaystyle {\mathbb {Q}}}.
  • The field E = K ( T ) {\displaystyle E=K(T)} {\displaystyle E=K(T)} of rational functions in one variable over an infinite field K {\displaystyle K} {\displaystyle K} is not a finitely generated algebra over K {\displaystyle K} {\displaystyle K}. On the other hand, E {\displaystyle E} {\displaystyle E} is generated over K {\displaystyle K} {\displaystyle K} by a single element, T {\displaystyle T} {\displaystyle T}, as a field.
  • If E / F {\displaystyle E/F} {\displaystyle E/F} is a finite field extension then it follows from the definitions that E {\displaystyle E} {\displaystyle E} is a finitely generated algebra over F {\displaystyle F} {\displaystyle F}.
  • Conversely, if E / F {\displaystyle E/F} {\displaystyle E/F} is a field extension and E {\displaystyle E} {\displaystyle E} is a finitely generated algebra over F {\displaystyle F} {\displaystyle F} then the field extension is finite. This is called Zariski's lemma. See also integral extension.
  • If G {\displaystyle G} {\displaystyle G} is a finitely generated group then the group algebra K G {\displaystyle KG} {\displaystyle KG} is a finitely generated algebra over K {\displaystyle K} {\displaystyle K}.
  • "Quadratic and quaternion algebras are essentially all the associative R-algebras which are finitely generated projective R-modules and have a standard involution."[2]

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 {\displaystyle A} {\displaystyle A} is a finitely generated commutative algebra over a Noetherian ring then every ideal of A is finitely generated, or equivalently, A {\displaystyle A} {\displaystyle 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}} {\displaystyle V\subseteq \mathbb {A} ^{n}} we can associate a finitely generated K {\displaystyle K} {\displaystyle K}-algebra

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

called the affine coordinate ring of V {\displaystyle V} {\displaystyle V}; moreover, if ϕ : V → W {\displaystyle \phi \colon V\to 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}} {\displaystyle V\subseteq \mathbb {A} ^{n}} and W ⊆ A m {\displaystyle W\subseteq \mathbb {A} ^{m}} {\displaystyle W\subseteq \mathbb {A} ^{m}}, we can define a homomorphism of K {\displaystyle 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 ,} {\displaystyle \Gamma (\phi )\equiv \phi ^{*}\colon \Gamma (W)\to \Gamma (V),\,\phi ^{*}(f)=f\circ \phi ,}

then, Γ {\displaystyle \Gamma } {\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} {\displaystyle K}-algebras: this functor turns out[3] 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}}),} {\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}}).} {\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} {\displaystyle R}-algebra A {\displaystyle A} {\displaystyle A} is a ring homomorphism ϕ : R → A {\displaystyle \phi \colon R\to A} {\displaystyle \phi \colon R\to A}; the R {\displaystyle R} {\displaystyle R}-module structure of A {\displaystyle 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.} {\displaystyle \lambda \cdot a:=\phi (\lambda )a,\quad \lambda \in R,a\in A.}

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

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

Again, there is a characterisation of finite algebras in terms of quotients:[4]

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

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

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

References

  1. Kemper, Gregor (2009). A Course in Commutative Algebra. Springer. p. 8. ISBN 978-3-642-03545-6.
  2. Max-Albert Knus (1991) Quadratic and Hermetian Forms over Rings, page 5, Grundlehren der Mathematischen Wissenschaften, Springer ISBN 3-540-52117-8 Zbl 0756.11008 doi:10.1007/978-3-642-75401-2
  3. 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.
  4. Atiyah, Michael Francis; Macdonald, Ian Grant (1994). Introduction to commutative algebra. CRC Press. p. 21. ISBN 9780201407518.

See also