Quasi-coherent sheaf
A sheaf of modules locally defined by generators and relations. More precisely, let 
be a topological space and let 
be a sheaf of rings on 
; a sheaf 
of 
-modules is called quasi-coherent if for any point 
there is an open neighbourhood 
and an exact sequence of sheaves of 
-modules
 |
where 
and 
are certain sets, 
denotes the restriction of a sheaf to 
and 
is the direct sum of 
copies of 
. A quasi-coherent sheaf is similarly defined on a topologized category with a sheaf of rings.
If 
is an affine scheme, then the association 
gives rise to an equivalence of the category of quasi-coherent sheaves of 
-modules and the category of 
-modules. As a result of this, quasi-coherent sheaves find broad application in the theory of schemes (see also Coherent sheaf; Scheme).
Comments
References
| [a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 111–115; 126 MR0463157 Zbl 0367.14001 |
Quasi-coherent sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-coherent_sheaf&oldid=48377