Definition:Category of Quasi-Coherent Sheaves of Modules

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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 $\map {\mathbf {QCoh} } {X, \OO_X}$.

Also see