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).
|[a1]||R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 111–115; 126|
Quasi-coherent sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-coherent_sheaf&oldid=13381