Base change theorems

In mathematics, the base change theorems relate the <a href="/facts/Direct_image_functor/z3mckxvp">direct image</a> and the <a href="/facts/Inverse_image_functor/wDG4ARH0">inverse image</a> of <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaves</a>. More precisely, they are about the base change map, given by the following <a href="/facts/Natural_transformation/xwPumTHt">natural transformation</a> of sheaves:

g
          
            ∗
          
        
        (
        
          R
          
            r
          
        
        
          f
          
            ∗
          
        
        
          
            F
          
        
        )
        →
        
          R
          
            r
          
        
        
          f
          
            ∗
          
          ′
        
        (
        
          g
          
            ′
            
              ∗
            
          
        
        
          
            F
          
        
        )
      
    
    {\displaystyle g^{*}(R^{r}f_{*}{\mathcal {F}})\to R^{r}f'_{*}(g'^{*}{\mathcal {F}})}

where

X
                  ′
                
              
              
                
                  
                    
                      
                        →
                      
                      
                        
                          g
                          ′
                        
                      
                    
                  
                
              
              
                X
              
            
            
              
                
                  f
                  ′
                
                ↓
              
              
              
                ↓
                f
              
            
            
              
                
                  S
                  ′
                
              
              
                
                  
                    
                      
                        →
                      
                      
                        g
                      
                    
                  
                
              
              
                S
              
            
          
        
      
    
    {\displaystyle {\begin{array}{rcl}X'&{\stackrel {g'}{\to }}&X\\f'\downarrow &&\downarrow f\\S'&{\stackrel {g}{\to }}&S\end{array}}}

is a <a href="/facts/Cartesian_square_(category_theory)/bC0QoX5U">Cartesian square</a> of topological spaces and 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 is a sheaf on X.
Such theorems exist in different branches of geometry: for (essentially arbitrary) topological spaces and <a href="/facts/Proper_map/jJS4YK7o">proper maps</a> f, in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a> for (quasi-)coherent sheaves and f proper or g flat, similarly in <a href="/facts/Analytic_geometry/3qWP0kM0">analytic geometry</a>, but also for <a href="/facts/%25C3%2589tale_sheaf/U67cR3JU">étale sheaves</a> for f proper or g smooth.

Base change theorems open-in-new

Base change theorems