Module theory MOC

Projective module

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

https://q.uiver.app/#q=WzAsMyxbMiwwLCJBIl0sWzIsMiwiQiJdLFswLDIsIlAiXSxbMCwxLCJxIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzIsMSwiZiIsMl0sWzIsMCwiXFxleGlzdHMgXFxiYXIgZiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==

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.

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Footnotes

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