View source for Quasi-coherent sheaf

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A [[Sheaf|sheaf]] of modules locally defined by generators and relations. More precisely, let  $  X $
be a topological space and let  $  {\mathcal A} $
be a sheaf of rings on  $  X $; 
a sheaf  $  {\mathcal F} $
of  $  {\mathcal A} $-
modules is called quasi-coherent if for any point  $  x \in X $
there is an open neighbourhood  $  U $
and an exact sequence of sheaves of  $  ( {\mathcal A} \mid  _ {U} ) $-
modules

$$ 
{\mathcal A} | _ {U}  ^ {(} I)  \rightarrow  {\mathcal A} | _ {U}  ^ {(} J)  \rightarrow \ 
{\mathcal F} \mid  _ {U}  \rightarrow  0 ,
$$

where  $  I $
and  $  J $
are certain sets,  $  \mid  _ {U} $
denotes the restriction of a sheaf to  $  U $
and  $  {\mathcal A}  ^ {(} I) $
is the direct sum of  $  I $
copies of  $  {\mathcal A} $. 
A quasi-coherent sheaf is similarly defined on a [[Topologized category|topologized category]] with a sheaf of rings.

If  $  ( X , {\mathcal A} ) $
is an affine scheme, then the association  $  {\mathcal F} \mapsto \Gamma ( X , {\mathcal F} ) $
gives rise to an equivalence of the category of quasi-coherent sheaves of  $  {\mathcal A} $-
modules and the category of  $  \Gamma ( X , {\mathcal A} ) $-
modules. As a result of this, quasi-coherent sheaves find broad application in the theory of schemes (see also [[Coherent sheaf|Coherent sheaf]]; [[Scheme|Scheme]]).

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 111–115; 126 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>