Sheaf generated by global sections | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-06-13

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Sheaf_of_modules


00:01:53 1 Examples
00:02:56 2 Operations
00:11:06 3 Properties
00:12:55 4 Sheaf associated to a module
00:19:40 5 Sheaf associated to a graded module
00:22:44 6 Computing sheaf cohomology
00:23:59 7 Sheaf extension
00:27:17 7.1 Locally Free Resolutions
00:29:19 7.2 Examples
00:29:28 7.2.1 Hypersurface
00:30:42 7.2.2 Union of Smooth Complete Intersections
00:35:55 8 See also



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SUMMARY
=======
In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) →F(V) are compatible with the restriction maps O(U) →O(V): the restriction of fs is the restriction of f times that of s for any f in O(U) and s in F(U).
The standard case is when X is a scheme and O its structure sheaf. If O is the constant sheaf 
  
    
      
        
          
            
              Z
            
            _
          
        
      
    
    {\displaystyle {\underline {\mathbf {Z} }}}
  , then a sheaf of O-modules is the same as a sheaf of abelian groups (i.e., an abelian sheaf).
If X is the prime spectrum of a ring R, then any R-module defines an OX-module (called an associated sheaf) in a natural way. Similarly, if R is a graded ring and X is the Proj of R, then any graded module defines an OX-module in a natural way. O-modules arising in such a fashion are examples of quasi-coherent sheaves, and in fact, on affine or projective schemes, all quasi-coherent sheaves are obtained this way.
Sheaves of modules over a ringed space form an abelian category. Moreover, this category has enough injectives, and consequently one can and does define the sheaf cohomology 
  
    
      
        
          H
          
            i
          
        
        ⁡
        (
        X
        ,
        −
        )
      
    
    {\displaystyle \operatorname {H} ^{i}(X,-)}
   as the i-th right derived functor of the global section functor 
  
    
      
        Γ
        (
        X
        ,
        −
        )
      
    
    {\displaystyle \Gamma (X,-)}
  .