§ Projective modules in terms of universal property

§ (1): Universal property / Defn

     e
   E ->> B
   ^   ^
  f~\  | f
     \ |
       P

§ Thm: every free module is projective

§ (1 => 2): Projective as splitting of exact sequences

            pi
0 -> N -> M -> P -> 0
               ^
               | idP
               P
            pi
0 -> N -> M -> P -> 0
          ^   ^
       idP~\  | idP
            \ |
             P

§ (2 => 3): Projective as direct summand of free module

§ Splitting lemma

§ (3 => 1): Direct summand of free module implies lifting

  e
E ->>B
     ^
    f|
     P
  e
E ->>B
     ^
    f| 
     P <<-- P(+)Q
         pi
  e
E ->>B <--
     ^    \f~
    f|     \
     P <<-- P(+)Q
         pi
--------g~--------
|                |
v e              |
E ->>B <--       g~
     ^    \f~    |
    f|     \     |
     P <<-- P(+)Q
         pi

§ Non example of projective module

§ Example of module that is projective but not free

§ References