Group-scheme action | Wikipedia audio article
Channel: wikipedia tts
Date: 29/03/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Group-scheme_action


00:02:49 1 Constructs
00:05:01 2 Problem of constructing a quotient
00:06:53 3 See also



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SUMMARY
=======
In algebraic geometry, a action of a group scheme is a generalization of a group action to a group scheme. Precisely, given a group S-scheme G, a left action of G on an S-scheme X is an S-morphism

  
    
      
        σ
        :
        G
        
          ×
          
            S
          
        
        X
        →
        X
      
    
    {\displaystyle \sigma :G\times _{S}X\to X}
  such that

(associativity) 
  
    
      
        σ
        ∘
        (
        
          1
          
            G
          
        
        ×
        σ
        )
        =
        σ
        ∘
        (
        m
        ×
        
          1
          
            X
          
        
        )
      
    
    {\displaystyle \sigma \circ (1_{G}\times \sigma )=\sigma \circ (m\times 1_{X})}
  , where 
  
    
      
        m
        :
        G
        
          ×
          
            S
          
        
        G
        →
        G
      
    
    {\displaystyle m:G\times _{S}G\to G}
   is the group law,
(unitality) 
  
    
      
        σ
        ∘
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        e
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          1
          
            X
          
        
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        =
        
          1
          
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    {\displaystyle \sigma \circ (e\times 1_{X})=1_{X}}
  , where 
  
    
      
        e
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        S
        →
        G
      
    
    {\displaystyle e:S\to G}
   is the identity section of G.A right action of G on X is defined analogously. A scheme equipped with a left or right action of a group scheme G is called a G-scheme. An equivariant morphism between G-schemes is a morphism of schemes that intertwines the respective G-actions.
More generally, one can also consider (at least some special case of) an action of a group functor: viewing G as a functor, an action is given as a natural transformation satisfying the conditions analogous to the above. Alternatively, some authors study group action in the language of a groupoid; a group-scheme action is then an example of a groupoid scheme.