# Quasi-coherent sheaf

A 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 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; Scheme).