Module theory MOC

Projective module

An -module is said to be projective iff it is a projective object in Category of left modules, i.e. for any morphism and epimorphism we have a lift.

This is equivalent to any of the following¹

1. preserves epimorphisms;
2. Any Module epimorphism splits;
3. is a direct summand of a free module, i.e. for some module and some cardinal ;
4. is exact.

Proof

If ^P1 holds, then taking and gives ^P2.

If ^P2 holds, consider an epimorphism . Then the split short exact sequence

guarantees the required direct sum decomposition, giving ^P3.

Note that is already exact if , so since

it follows from ^P1 that ^P3 implies ^P4.

Noting that being a module epimorphism is the same as being a regular epimorphism, and that the latter must be preserved by exact functors, it is clear that ^P4 implies ^P3.

---

develop | en | sembr

Footnotes

1. 2011. Introduction to representation theory, §8.1, pp. 205–332 ↩