over the ring for every open set in , and the restriction maps are compatible with the module structure, then we call an -module. In this case, the stalk of at will be a module over the local ring (stalk) , for every . A morphism between two such -modules is a morphism of sheaves which is compatible with the given module structures. The category of -modules over a fixed locally ringed space is an abelian category. An important subcategory of the category of -modules is the category of quasi-coherent sheaves on . A sheaf of -modules is called quasi-coherent if it is, locally, isomorphic to the cokernel of a map between free -modules. A coherent sheaf is a quasi-coherent sheaf which is, locally, of finite type and for every open subset of the kernel of any morphism from a free -modules of finite rank to is also of finite type. Citations References Section 0.4 of External links Sheaf theory Scheme theory