Projection formula

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, the projection formula states the following:
For a morphism 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 of <a href="/facts/Ringed_space/JyeaTMtX">ringed spaces</a>, an 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-module 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 and a <a href="/facts/Locally_free_sheaf/h2NoUN7e">locally free</a> 
 
 
 
 
 
 
 O
 
 
 
 Y
 
 
 
 
 {\displaystyle {\mathcal {O}}_{Y}}
 
-module 
 
 
 
 
 
 E
 
 
 
 
 {\displaystyle {\mathcal {E}}}
 
 of finite rank, the natural maps of sheaves

R
          
            i
          
        
        
          f
          
            ∗
          
        
        
          
            F
          
        
        ⊗
        
          
            E
          
        
        →
        
          R
          
            i
          
        
        
          f
          
            ∗
          
        
        (
        
          
            F
          
        
        ⊗
        
          f
          
            ∗
          
        
        
          
            E
          
        
        )
      
    
    {\displaystyle R^{i}f_{*}{\mathcal {F}}\otimes {\mathcal {E}}\to R^{i}f_{*}({\mathcal {F}}\otimes f^{*}{\mathcal {E}})}

are isomorphisms.
There is yet another projection formula in the setting of <a href="/facts/%25C3%2589tale_cohomology/KXJ8a6Hw">étale cohomology</a>.

Projection formula open-in-new

Projection formula