Group-scheme action

<h2 id="constructs">Constructs</h2>
<p>The usual constructs for a <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group action</a> such as orbits generalize to a group-scheme action. Let 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
 be a given group-scheme action as above.
</p>
<ul><li>Given a T-valued point 
  
    
      
        x
        :
        T
        →
        X
      
    
    {\displaystyle x:T\to X}
  
, the orbit map 
  
    
      
        
          σ
          
            x
          
        
        :
        G
        
          ×
          
            S
          
        
        T
        →
        X
        
          ×
          
            S
          
        
        T
      
    
    {\displaystyle \sigma _{x}:G\times _{S}T\to X\times _{S}T}
  
 is given as 
  
    
      
        (
        σ
        ∘
        (
        
          1
          
            G
          
        
        ×
        x
        )
        ,
        
          p
          
            2
          
        
        )
      
    
    {\displaystyle (\sigma \circ (1_{G}\times x),p_{2})}
  
.</li>
<li>The <a href="/facts/Orbit_(group_theory)/Vh9VtUUw">orbit</a> of <i>x</i> is the image of the orbit map 
  
    
      
        
          σ
          
            x
          
        
      
    
    {\displaystyle \sigma _{x}}
  
.</li>
<li>The <a href="/facts/Stabilizer_subgroup/Vh9VtUUw">stabilizer</a> of <i>x</i> is the <a href="/facts/Scheme-theoretic_fiber/gSFuGC7l">fiber</a> over 
  
    
      
        
          σ
          
            x
          
        
      
    
    {\displaystyle \sigma _{x}}
  
 of the map 
  
    
      
        (
        x
        ,
        
          1
          
            T
          
        
        )
        :
        T
        →
        X
        
          ×
          
            S
          
        
        T
        .
      
    
    {\displaystyle (x,1_{T}):T\to X\times _{S}T.}
  
</li></ul>
<h2 id="problem-of-constructing-a-quotient">Problem of constructing a quotient</h2>

<p>Unlike a set-theoretic group action, there is no straightforward way to construct a quotient for a group-scheme action. One exception is the case when the action is free, the case of a <a href="/facts/Principal_fiber_bundle/bbB2D94B">principal fiber bundle</a>.
</p><p>There are several approaches to overcome this difficulty:
</p>
<ul><li><a href="/facts/Level_structure_(algebraic_geometry)/aJBArxJa">Level structure</a> - Perhaps the oldest, the approach replaces an object to classify by an object together with a level structure</li>
<li><a href="/facts/Geometric_invariant_theory/l4eV0RdG">Geometric invariant theory</a> - throw away bad orbits and then take a quotient. The drawback is that there is no canonical way to introduce the notion of "bad orbits"; the notion depends on a choice of <a href="/facts/Linearization/ohYSeQR3">linearization</a>. See also: <a href="/facts/Categorical_quotient/0KTb8W6M">categorical quotient</a>, <a href="/facts/GIT_quotient/BGS7ASbx">GIT quotient</a>.</li>
<li><a href="/facts/Borel_construction/CvsVQcg6">Borel construction</a> - this is an approach essentially from algebraic topology; this approach requires one to work with an <a href="/facts/Infinite-dimensional_space/z8m02A3W">infinite-dimensional space</a>.</li>
<li>Analytic approach, the theory of <a href="/facts/Teichm%C3%BCller_space/IjwtcRyb">Teichmüller space</a></li>
<li><a href="/facts/Quotient_stack/lwme8cqw">Quotient stack</a> - in a sense, this is the ultimate answer to the problem. Roughly, a "quotient prestack" is the category of orbits and one <a href="/facts/Stackification/2byjNpEA">stackify</a> (i.e., the introduction of the notion of a torsor) it to get a quotient stack.</li></ul>
<p>Depending on applications, another approach would be to shift the focus away from a space then onto stuff on a space; e.g., <a href="/facts/Topos/zzttCIwC">topos</a>. So the problem shifts from the classification of orbits to that of equivariant objects.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Groupoid_scheme/x3pXQJLx">groupoid scheme</a></li>
<li><a href="/facts/Sumihiro%27s_theorem/keYsXJG9">Sumihiro's theorem</a></li>
<li><a href="/facts/Equivariant_sheaf/sAItdnmQ">equivariant sheaf</a></li>
<li><a href="/facts/Borel_fixed-point_theorem/LqXxKRmz">Borel fixed-point theorem</a></li></ul>

<ul><li><a href="/facts/David_Mumford/WJoJ8K2u">Mumford, David</a>; Fogarty, J.; <a href="/facts/Frances_Kirwan/puzY653u">Kirwan, F.</a> (1994). <i>Geometric invariant theory</i>. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Vol. 34 (3rd ed.). Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-56963-3. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1304906">1304906</a>.</li></ul>

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

<ol>
<li id="fn:1"><p>In details, given a group-scheme action 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
, for each morphism 
  
    
      
        T
        →
        S
      
    
    {\displaystyle T\to S}
  
, 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
 determines a group action 
  
    
      
        G
        (
        T
        )
        ×
        X
        (
        T
        )
        →
        X
        (
        T
        )
      
    
    {\displaystyle G(T)\times X(T)\to X(T)}
  
; i.e., the group 
  
    
      
        G
        (
        T
        )
      
    
    {\displaystyle G(T)}
  
 acts on the set of T-points 
  
    
      
        X
        (
        T
        )
      
    
    {\displaystyle X(T)}
  
. Conversely, if for each 
  
    
      
        T
        →
        S
      
    
    {\displaystyle T\to S}
  
, there is a group action 
  
    
      
        
          σ
          
            T
          
        
        :
        G
        (
        T
        )
        ×
        X
        (
        T
        )
        →
        X
        (
        T
        )
      
    
    {\displaystyle \sigma _{T}:G(T)\times X(T)\to X(T)}
  
 and if those actions are compatible; i.e., they form a natural transformation, then, by the Yoneda lemma, they determine a group-scheme action 
  
    
      
        σ
        :
        G
        
          ×
          
            S
          
        
        X
        →
        X
      
    
    {\displaystyle \sigma :G\times _{S}X\to X}
  
. <a href="/wiki/Natural_transformation" target="_blank">/wiki/Natural_transformation</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Group-scheme action open-in-new

Group-scheme action