Ringed Space - O_X Modules

O_X Modules

Given a locally ringed space (X, O_X), certain sheaves of modules on X occur in the applications, the O_X-modules. To define them, consider a sheaf F of abelian groups on X. If F(U) is a module over the ring O_X(U) for every open set U in X, and the restriction maps are compatible with the module structure, then we call F an O_X-module. In this case, the stalk of F at x will be a module over the local ring (stalk) R_x, for every x∈X.

A morphism between two such O_X-modules is a morphism of sheaves which is compatible with the given module structures. The category of O_X-modules over a fixed locally ringed space (X, O_X) is an abelian category.

An important subcategory of the category of O_X-modules is the category of quasi-coherent sheaves on X. A sheaf of O_X-modules is called quasi-coherent if it is, locally, isomorphic to the cokernel of a map between free O_X-modules. A coherent sheaf F is a quasi-coherent sheaf which is, locally, of finite type and for every open subset U of X the kernel of any morphism from a free O_U-modules of finite rank to F_U is also of finite type.