Group-scheme action

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, an action of a group scheme is a generalization of a <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group action</a> to a <a href="/facts/Group_scheme/bJW14FE2">group scheme</a>. 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

<ul><li>(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,</li>
<li>(unitality) 
 
 
 
 σ
 ∘
 (
 e
 ×
 
 1
 
 X
 
 
 )
 =
 
 1
 
 X
 
 
 
 
 {\displaystyle \sigma \circ (e\times 1_{X})=1_{X}}
 
, where 
 
 
 
 e
 :
 S
 →
 G
 
 
 {\displaystyle e:S\to G}
 
 is the identity section of G.</li></ul>
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 <a href="/facts/Equivariant_morphism/KkIkDwIH">equivariant morphism</a> between G-schemes is a <a href="/facts/Morphism_of_schemes/55CKqHHo">morphism of schemes</a> that intertwines the respective G-actions.
More generally, one can also consider (at least some special case of) an action of a <a href="/facts/Group_functor/Unor4H35">group functor</a>: 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 <a href="/facts/Groupoid/hvj8wvWe">groupoid</a>; a group-scheme action is then an example of a <a href="/facts/Groupoid_scheme/x3pXQJLx">groupoid scheme</a>.

Group-scheme action open-in-new

Group-scheme action