Definition:Category of Quasi-Coherent Sheaves of Modules

Definition
Let $X$ be a topological space.

Let $\OO_X$ be a sheaf of commutative rings.

The category of quasi-coherent $\OO_X$-modules $\map {\mathbf {QCoh} } {X, \OO_X}$ is the category with:

Also denoted as
If $\struct {X, \OO_X}$ is a scheme, one also writes $\map {\mathbf {QCoh} } X$ instead of $\map {\mathbf {QCoh} } {X, \OO_X}$.

Also see

 * Definition:Category of Presheaves on Topological Space
 * Definition:Ringed Space