Projective module

An $𝑅$ -module $𝑃$ is said to be projective iff it is a projective object in $𝑅 𝖬 𝗈 𝖽$ , i.e. for any morphism $𝑓 : 𝑃 \to 𝐵$ and epimorphism $𝑞 : 𝐴 ↠ 𝐵$ we have a lift.

This is equivalent to any of the following¹

$𝑅 𝖬 𝗈 𝖽 (𝑃, -)$ preserves epimorphisms;
Any Module epimorphism $𝛼 : 𝑀 ↠ 𝑃$ splits;
$𝑃$ is a direct summand of a free module, i.e. $𝑃 \oplus 𝑄 ≅ 𝑅 (𝛼)$ for some module $𝑄$ and some cardinal $𝛼$ ;
$𝑅 𝖬 𝗈 𝖽 (𝑃, -)$ is exact.

Proof

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

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

0 \to k e r 𝑝 ↪ 𝑅 (𝛼) ↠ 𝑃 \to 0

guarantees the required direct sum decomposition, giving ^P3.

Note that $𝑅 𝖬 𝗈 𝖽 (𝑃, -)$ is already exact if $𝑃 = 𝑅$ , so since

𝑅 𝖬 𝗈 𝖽 (∐ 𝑖 \in 𝐼 𝑀 𝑖, 𝑁) ≅ ∐ 𝑖 \in 𝐼 𝑅 𝖬 𝗈 𝖽 (𝑀 𝑖, 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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2011. Introduction to representation theory, §8.1, pp. 205–332 ↩

Projective module

Footnotes