Tangent space to a functor | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-03

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SUMMARY
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In algebraic geometry, the tangent space to a functor generalizes the classical construction of a tangent space such as the Zariski tangent space. The construction is based on the following observation. Let X be a scheme over a field k.

To give a 
  
    
      
        k
        [
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          /
        
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        ϵ
        
          )
          
            2
          
        
      
    
    {\displaystyle k[\epsilon ]/(\epsilon )^{2}}
  -point of X is the same thing as to give a k-rational point p of X (i.e., the residue field of p is k) together with an element of 
  
    
      
        (
        
          
            
              m
            
          
          
            X
            ,
            p
          
        
        
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              m
            
          
          
            X
            ,
            p
          
          
            2
          
        
        
          )
          
            ∗
          
        
      
    
    {\displaystyle ({\mathfrak {m}}_{X,p}/{\mathfrak {m}}_{X,p}^{2})^{*}}
  ; i.e., a tangent vector at p.(To see this, use the fact that any local homomorphism 
  
    
      
        
          
            
              O
            
          
          
            p
          
        
        →
        k
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          /
        
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            2
          
        
      
    
    {\displaystyle {\mathcal {O}}_{p}\to k[\epsilon ]/(\epsilon )^{2}}
   must be of the form

  
    
      
        
          δ
          
            p
          
          
            v
          
        
        :
        u
        ↦
        u
        (
        p
        )
        +
        ϵ
        v
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        u
        )
        ,
        
        v
        ∈
        
          
            
              O
            
          
          
            p
          
          
            ∗
          
        
        .
      
    
    {\displaystyle \delta _{p}^{v}:u\mapsto u(p)+\epsilon v(u),\quad v\in {\mathcal {O}}_{p}^{*}.}
  )Let F be a functor from the category of k-algebras to the category of sets. Then, for any k-point 
  
    
      
        p
        ∈
        F
        (
        k
        )
      
    
    {\displaystyle p\in F(k)}
  , the fiber of 
  
    
      
        π
        :
        F
        (
        k
        [
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        ]
        
          /
        
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            2
          
        
        )
        →
        F
        (
        k
        )
      
    
    {\displaystyle \pi :F(k[\epsilon ]/(\epsilon )^{2})\to F(k)}
   over p is called the tangent space to F at p.
The tangent space may be given the structure of a vector space over k. If F is a scheme X over k (i.e., 
  
    
      
        F
        =
        
          Hom
          
            Spec
            ⁡
            k
          
        
        ⁡
        (
        Spec
        −
        ,
        X
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    {\displaystyle F=\operatorname {Hom} _{\operatorname {Spec} k}(\operatorname {Spec} -,X)}
  ), then each v as above may be identified with a derivation at p and this gives the identification of 
  
    
      
        
          π
          
            −
            1
          
        
        (
        p
        )
      
    
    {\displaystyle \pi ^{-1}(p)}
   with the space of derivations at p and we recover the usual construction.
The construction may be thought of as defining an analog of the tangent bundle in the fo ...