Sheaf (mathematics) - Sheaves of Modules

Sheaves of Modules

Let be a ringed space. A sheaf of modules is a sheaf such that on every open set U of X, is an -module and for every inclusion of open sets V ⊆ U, the restriction map is a homomorphism of -modules.

Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence between vector bundles and locally free sheaves of -modules. Sheaves of solutions to differential equations are D-modules, that is, modules over the sheaf of differential operators.

A particularly important case are abelian sheaves, which are modules over the constant sheaf . Every sheaf of modules is an abelian sheaf.