Module theory MOC

Projective module

An 𝑅-module 𝑃 is said to be projective iff it is a projective object in Category of left modules, i.e. for any morphism 𝑓 :𝑃 →𝐵 and epimorphism 𝑞 :𝐴 ↠𝐵 we have a lift.

This is equivalent to any of the following¹

1. 𝑅𝖬𝗈𝖽(𝑃, −) preserves epimorphisms;
2. Any Module epimorphism 𝛼 :𝑀 ↠𝑃 splits;
3. 𝑃 is a direct summand of a free module, i.e. 𝑃 ⊕𝑄 ≅𝑅(𝛼) for some module 𝑄 and some cardinal 𝛼;
4. 𝑅𝖬𝗈𝖽(𝑃, −) is exact.

Proof

If ^P1 holds, then taking 𝐵 =𝑃 and 𝑓 =1𝑃 gives ^P2.

If ^P2 holds, consider an epimorphism 𝑝 :𝑅(𝛼) ↠𝑃. Then the split short exact sequence

0→ker⁡𝑝↪𝑅(𝛼)↠𝑃→0

guarantees the required direct sum decomposition, giving ^P3.

Note that 𝑅𝖬𝗈𝖽(𝑃, −) is already exact if 𝑃 =𝑅, so since

𝑅𝖬𝗈𝖽(∐𝑖∈𝐼𝑀𝑖,𝑁)≅∐𝑖∈𝐼𝑅𝖬𝗈𝖽(𝑀𝑖,𝑁)

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 ↩