Finitely presented module

Summary

A finitely presented module is a module given by finitely many generators and finitely many relations among them. It is the cokernel of a linear map between two finitely generated free modules, so it is described by a finite matrix. Every finitely presented module is finitely generated, and over a Noetherian ring the two notions agree. Many statements about modules, such as the behaviour of $\Hom$ under localization, need finite presentation rather than finite generation.