Artin approximation theorem

<h2 id="statement-of-the-theorem">Statement of the theorem</h2>
<p>Let 
  
    
      
        
          x
        
        =
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
      
    
    {\displaystyle \mathbf {x} =x_{1},\dots ,x_{n}}
  
 denote a collection of <i>n</i> <a href="/facts/Indeterminate_(variable)/R7QAHS4j">indeterminates</a>, 
  
    
      
        k
        [
        [
        
          x
        
        ]
        ]
      
    
    {\displaystyle k[[\mathbf {x} ]]}
  
  the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> of formal <a href="/facts/Power_series/vYh6sTps">power series</a> with indeterminates 
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
 over a field <i>k</i>, and 
  
    
      
        
          y
        
        =
        
          y
          
            1
          
        
        ,
        …
        ,
        
          y
          
            n
          
        
      
    
    {\displaystyle \mathbf {y} =y_{1},\dots ,y_{n}}
  
 a different set of indeterminates. Let 
</p>

f
        (
        
          x
        
        ,
        
          y
        
        )
        =
        0
      
    
    {\displaystyle f(\mathbf {x} ,\mathbf {y} )=0}

<p>be a system of <a href="/facts/Polynomial_equation/1QauIGxs">polynomial equations</a> in 
  
    
      
        k
        [
        
          x
        
        ,
        
          y
        
        ]
      
    
    {\displaystyle k[\mathbf {x} ,\mathbf {y} ]}
  
, and <i>c</i> a positive <a href="/facts/Integer/PUwwrawx">integer</a>. Then given a formal power series solution 
  
    
      
        
          
            
              
                y
              
              ^
            
          
        
        (
        
          x
        
        )
        ∈
        k
        [
        [
        
          x
        
        ]
        ]
      
    
    {\displaystyle {\hat {\mathbf {y} }}(\mathbf {x} )\in k[[\mathbf {x} ]]}
  
, there is an algebraic solution 
  
    
      
        
          y
        
        (
        
          x
        
        )
      
    
    {\displaystyle \mathbf {y} (\mathbf {x} )}
  
 consisting of <a href="/facts/Algebraic_function/yT7EKmwF">algebraic functions</a> (more precisely, algebraic power series) such that 
</p>

y
              
              ^
            
          
        
        (
        
          x
        
        )
        ≡
        
          y
        
        (
        
          x
        
        )
        
          mod
          
            (
          
        
        
          x
        
        
          )
          
            c
          
        
        .
      
    
    {\displaystyle {\hat {\mathbf {y} }}(\mathbf {x} )\equiv \mathbf {y} (\mathbf {x} ){\bmod {(}}\mathbf {x} )^{c}.}

<h2 id="discussion">Discussion</h2>
<p>Given any desired positive integer <i>c</i>, this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by <i>c</i>. This leads to theorems that deduce the existence of certain <a href="/facts/Formal_moduli_space/HM7EuH3t">formal moduli spaces</a> of deformations as <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a>. See also: <a href="/facts/Artin%27s_criterion/mslWTvFs">Artin's criterion</a>.
</p>
<h2 id="alternative-statement">Alternative statement</h2>
<p>The following alternative statement is given in Theorem 1.12 of <a href="/facts/Michael_Artin/Tfk0f6rK">Michael Artin</a> (1969).
</p><p>Let 
  
    
      
        R
      
    
    {\displaystyle R}
  
 be a field or an excellent discrete valuation ring, let 
  
    
      
        A
      
    
    {\displaystyle A}
  
 be the <a href="/facts/Henselian_ring/rJaECl4s">henselization</a> at a prime ideal of an 
  
    
      
        R
      
    
    {\displaystyle R}
  
-algebra of finite type, let <i>m</i> be a proper ideal of 
  
    
      
        A
      
    
    {\displaystyle A}
  
, let 
  
    
      
        
          
            
              A
              ^
            
          
        
      
    
    {\displaystyle {\hat {A}}}
  
 be the <i>m</i>-adic completion of 
  
    
      
        A
      
    
    {\displaystyle A}
  
, and let 
</p>

F
        :
        (
        A
        
          -algebras
        
        )
        →
        (
        
          sets
        
        )
        ,
      
    
    {\displaystyle F\colon (A{\text{-algebras}})\to ({\text{sets}}),}

<p>be a functor sending filtered colimits to filtered colimits (Artin calls such a functor locally of finite presentation).  Then for any integer <i>c</i> and any 
  
    
      
        
          
            ξ
            ¯
          
        
        ∈
        F
        (
        
          
            
              A
              ^
            
          
        
        )
      
    
    {\displaystyle {\overline {\xi }}\in F({\hat {A}})}
  
, there is a 
  
    
      
        ξ
        ∈
        F
        (
        A
        )
      
    
    {\displaystyle \xi \in F(A)}
  
 such that 
</p>

ξ
            ¯
          
        
        ≡
        ξ
        
          
            mod
            
              m
            
          
          
            c
          
        
      
    
    {\displaystyle {\overline {\xi }}\equiv \xi {\bmod {m}}^{c}}
  
.
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Ring_with_the_approximation_property/NA5r2mLz">Ring with the approximation property</a></li>
<li><a href="/facts/Popescu%27s_theorem/pnVY4nDS">Popescu's theorem</a></li>
<li><a href="/facts/Artin%27s_criterion/mslWTvFs">Artin's criterion</a></li></ul>

<ul><li><a href="/facts/Michael_Artin/Tfk0f6rK">Artin, Michael</a> (1969), <a href="http://www.numdam.org/item?id=PMIHES_1969__36__23_0">"Algebraic approximation of structures over complete local rings"</a>, <i><a href="/facts/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S/2ur5ofxL">Publications Mathématiques de l'IHÉS</a></i>, 36 (36): 23–58, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02684596">10.1007/BF02684596</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0268188">0268188</a></li>
<li>Artin, Michael (1971). <i>Algebraic Spaces</i>. Yale Mathematical Monographs. Vol. 3. New Haven, CT–London: <a href="/facts/Yale_University_Press/oavMPaT7">Yale University Press</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0407012">0407012</a>.</li>
<li><a href="/facts/Michel_Raynaud/zvNqbhK4">Raynaud, Michel</a> (1971), <a href="http://www.numdam.org/book-part/SB_1968-1969__11__279_0/">"Travaux récents de M. Artin"</a>, <i><a href="/facts/S%C3%A9minaire_Nicolas_Bourbaki/uc2ldL81">Séminaire Nicolas Bourbaki</a></i>, 11 (363): 279–295, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3077132">3077132</a></li></ul>

Artin approximation theorem open-in-new

Artin approximation theorem