Finitely-generated Module - Finitely Presented, Finitely Related, and Coherent Modules

Finitely Presented, Finitely Related, and Coherent Modules

Another formulation is this: a finitely generated module M is one for which there is a epimorphism

f : Rk → M.

Suppose now there is an epimorphism,

φ : F → M.

for a module M and free module F.

• If the kernel of φ is finitely generated, then M is called a finitely related module. Since M is isomorphic to F/ker(φ), this basically expresses that M is obtained by taking a free module and introducing finitely many relations within F (the generators of ker(φ)).
• If the kernel of φ is finitely generated and F has finite rank (i.e. F=Rk), then M is said to be a finitely presented module. Here, M is specified using finitely many generators (the images of the k generators of F=Rk) and finitely many relations (the generators of ker(φ)).
• A coherent module M is a finitely generated module whose finitely generated submodules are finitely presented.

Over any ring R, coherent modules are finitely presented, and finitely presented modules are both finitely generated and finitely related. For a Noetherian ring R, all four conditions are actually equivalent.

Some crossover occurs for projective or flat modules. A finitely generated projective module is finitely presented, and a finitely related flat module is projective.

It is true also that the following conditions are equivalent for a ring R:

1. R is a right coherent ring.
2. The module R_R is a coherent module.
3. Every finitely presented right R module is coherent.

Although coherence seems like a more cumbersome condition than finitely generated or finitely presented, it is nicer than them since the category of coherent modules is an abelian category, while, in general, neither finitely generated nor finitely presented modules form an abelian category.