Finitely generated algebra

<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Polynomial_algebra/ua3vk8Ih">polynomial algebra</a> 
  
    
      
        K
        [
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        ]
      
    
    {\displaystyle K[x_{1},\dots ,x_{n}]}
  
 is finitely generated. The polynomial algebra in <a href="/facts/Countable/Wj4DmWPv">countably infinitely many</a> generators is infinitely generated.</li>
<li>The field 
  
    
      
        E
        =
        K
        (
        t
        )
      
    
    {\displaystyle E=K(t)}
  
 of <a href="/facts/Rational_function/2nJGu0Ko">rational functions</a> in one variable over an infinite field <i>
  
    
      
        K
      
    
    {\displaystyle K}
  
</i> is <i>not</i> a finitely generated algebra over <i>
  
    
      
        K
      
    
    {\displaystyle K}
  
</i>. On the other hand, 
  
    
      
        E
      
    
    {\displaystyle E}
  
 is generated over 
  
    
      
        K
      
    
    {\displaystyle K}
  
 by a single element, <i>
  
    
      
        t
      
    
    {\displaystyle t}
  
</i>, <i>as a field</i>.</li>
<li>If 
  
    
      
        E
        
          /
        
        F
      
    
    {\displaystyle E/F}
  
 is a <a href="/facts/Finite_field_extension/L7yIDtyK">finite field extension</a> then it follows from the definitions that 
  
    
      
        E
      
    
    {\displaystyle E}
  
 is a finitely generated algebra over 
  
    
      
        F
      
    
    {\displaystyle F}
  
.</li>
<li>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 <a href="/facts/Zariski%2527s_lemma/B87rvzNK">Zariski's lemma</a>. See also <a href="/facts/Integral_extension/55dAuE39">integral extension</a>.</li>
<li>If 
  
    
      
        G
      
    
    {\displaystyle G}
  
 is a <a href="/facts/Finitely_generated_group/fVsFHrmb">finitely generated group</a> then the <a href="/facts/Group_ring/TJlkt5Te">group algebra</a> 
  
    
      
        K
        G
      
    
    {\displaystyle KG}
  
 is a finitely generated algebra over 
  
    
      
        K
      
    
    {\displaystyle K}
  
.</li></ul>
<h2 id="properties">Properties</h2>
<ul><li>A <a href="/facts/Homomorphism/0gyqdEse">homomorphic</a> <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of a finitely generated algebra is itself finitely generated. However,  a similar property for <a href="/facts/Subalgebra/9PEHNF5r">subalgebras</a> does not hold in general.</li>
<li><a href="/facts/Hilbert%2527s_basis_theorem/6Pman9ee">Hilbert's basis theorem</a>: if <i>A</i> is a finitely generated commutative algebra over a <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian ring</a> then every <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> of <i>A</i> is finitely generated, or equivalently, <i>A</i> is a Noetherian ring.</li></ul>
<h2 id="relation-with-affine-varieties">Relation with affine varieties</h2>
<p>Finitely generated <a href="/facts/Reduced_ring/Mjbr3zxj">reduced</a> <a href="/facts/Commutative_algebra_(structure)/DRfoF6KV">commutative algebras</a> are basic objects of consideration in modern <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, where they correspond to <a href="/facts/Affine_algebraic_variety/WVynz2Kn">affine algebraic varieties</a>; 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
</p>

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

<p>called the affine <a href="/facts/Coordinate_ring/WVynz2Kn">coordinate ring</a> of 
  
    
      
        V
      
    
    {\displaystyle V}
  
; moreover, if 
  
    
      
        ϕ
        :
        V
        →
        W
      
    
    {\displaystyle \phi \colon V\to W}
  
 is a <a href="/facts/Regular_map_(algebraic_geometry)/5jBzIgps">regular map</a> 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
</p>

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 ,}

<p>then, 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 is a <a href="/facts/Contravariant_functor/l2u3rIBE">contravariant functor</a> from the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of affine algebraic sets with regular maps to the category of reduced finitely generated 
  
    
      
        K
      
    
    {\displaystyle K}
  
-algebras: this functor turns out<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> to be an <a href="/facts/Equivalence_of_categories/k6uzqAcy">equivalence of categories</a>
</p>

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}}),}

<p>and, restricting to <a href="/facts/Affine_variety/WVynz2Kn">affine varieties</a> (i.e. <a href="/facts/Irreducible_space/Yu2eFIon">irreducible</a> affine algebraic sets),
</p>

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}}).}

<h2 id="finite-algebras-vs-algebras-of-finite-type">Finite algebras vs algebras of finite type</h2>
<p>We recall that a commutative 
  
    
      
        R
      
    
    {\displaystyle R}
  
-<a href="/facts/Associative_algebra/DRfoF6KV">algebra</a> 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is a <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphism</a> 
  
    
      
        ϕ
        :
        R
        →
        A
      
    
    {\displaystyle \phi \colon R\to A}
  
; the 
  
    
      
        R
      
    
    {\displaystyle R}
  
-<a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> structure of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is defined by
</p>

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

<p>An 
  
    
      
        R
      
    
    {\displaystyle R}
  
-algebra 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is called <i>finite</i> if it is <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> as an 
  
    
      
        R
      
    
    {\displaystyle R}
  
-module, i.e. there is a surjective homomorphism of 
  
    
      
        R
      
    
    {\displaystyle R}
  
-modules
</p>

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

<p>Again, there is a characterisation of <a href="/facts/Finite_algebra/mxlQykCW">finite algebras</a> in terms of quotients<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
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}
  
-<a href="/facts/Submodule/jP0LIhFL">submodule</a> 
  
    
      
        M
        ⊆
        R
      
    
    {\displaystyle M\subseteq R}
  
.
<p>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 <a href="/facts/Integral_element/55dAuE39">integral</a>, 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}
  
.
</p><p>Finite algebras and algebras of finite type are related to the notions of <a href="/facts/Finite_morphism/djrynAmJ">finite morphisms</a> and <a href="/facts/Finite_morphism/djrynAmJ">morphisms of finite type</a>.
</p>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Finitely_generated_module/6Njcpgzr">Finitely generated module</a></li>
<li><a href="/facts/Finitely_generated_field_extension/L7yIDtyK">Finitely generated field extension</a></li>
<li><a href="/facts/Artin%25E2%2580%2593Tate_lemma/nJhMPHqt">Artin–Tate lemma</a></li>
<li><a href="/facts/Finite_algebra/mxlQykCW">Finite algebra</a></li>
<li><a href="/facts/Morphism_of_finite_type/djrynAmJ">Morphism of finite type</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Kemper, Gregor (2009). A Course in Commutative Algebra. Springer. p. 8. ISBN 978-3-642-03545-6. <a href="978-3-642-03545-6" target="_blank">978-3-642-03545-6</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>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. <a href="978-3-8348-0676-5" target="_blank">978-3-8348-0676-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Atiyah, Michael Francis; Macdonald, Ian Grant (1994). Introduction to commutative algebra. CRC Press. p. 21. ISBN 9780201407518. <a href="9780201407518" target="_blank">9780201407518</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Finitely generated algebra open-in-new

Finitely generated algebra