Definition:Category of Quasi-Coherent Sheaves of Modules
Jump to navigation
Jump to search
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:
Objects: | quasi-coherent $\OO_X$-modules | |
Morphisms: | morphisms $\OO_X$-modules | |
Composition: | composition of morphisms of presheaves | |
Identity morphisms: | identity morphisms on presheaves |
Also denoted as
If $\struct {X, \OO_X}$ is a scheme, one also writes $\map {\mathbf {QCoh} } X$ instead of $\map {\mathbf {QCoh} } {X, \OO_X}$.