Equivariant sheaf

In mathematics, given an <a href="/facts/Group-scheme_action/jVo9ubJf">action</a> 
 
 
 
 σ
 :
 G
 
 ×
 
 S
 
 
 X
 →
 X
 
 
 {\displaystyle \sigma :G\times _{S}X\to X}
 
 of a <a href="/facts/Group_scheme/bJW14FE2">group scheme</a> G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of <a href="/facts/Sheaf_of_modules/grSMBXho">
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-modules</a> together with the isomorphism of 
 
 
 
 
 
 
 O
 
 
 
 G
 
 ×
 
 S
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{G\times _{S}X}}
 
-modules

ϕ
 :
 
 σ
 
 ∗
 
 
 F
 
 →
 
 ≃
 
 
 
 p
 
 2
 
 
 ∗
 
 
 F
 
 
 {\displaystyle \phi :\sigma ^{*}F\xrightarrow {\simeq } p_{2}^{*}F}
 
  
that satisfies the cocycle condition: writing m for multiplication,

p
          
            23
          
          
            ∗
          
        
        ϕ
        ∘
        (
        
          1
          
            G
          
        
        ×
        σ
        
          )
          
            ∗
          
        
        ϕ
        =
        (
        m
        ×
        
          1
          
            X
          
        
        
          )
          
            ∗
          
        
        ϕ
      
    
    {\displaystyle p_{23}^{*}\phi \circ (1_{G}\times \sigma )^{*}\phi =(m\times 1_{X})^{*}\phi }
  
.

Equivariant sheaf open-in-new

Equivariant sheaf