Editing Projective module (section)

== Definitions ==

=== Lifting property ===

The usual [[category theory|category theoretical]] definition is in terms of the property of [[lifting property|''lifting'']] that carries over from free to projective modules: a module ''P'' is projective [[if and only if]] for every [[surjective]] [[module homomorphism]] {{nowrap|''f'' : ''N'' ↠ ''M''}} and every module homomorphism {{nowrap|''g'' : ''P'' → ''M''}}, there exists a module homomorphism {{nowrap|''h'' : ''P'' → ''N''}} such that {{nowrap|1=''fh'' = ''g''}}. (We don't require the lifting homomorphism ''h'' to be unique; this is not a [[universal property]].)

:[[Image:Projective-module-P.svg|120px]]

The advantage of this definition of "projective" is that it can be carried out in [[category (mathematics)|categories]] more general than [[module categories]]: we don't need a notion of "free object". It can also be [[dual (category theory)|dualized]], leading to [[injective module]]s. The lifting property may also be rephrased as ''every morphism from <math>P</math> to <math>M</math> factors through every epimorphism to <math>M</math>''. Thus, by definition, projective modules are precisely the [[projective object]]s in the [[category of modules|category of ''R''-modules]].

=== Split-exact sequences ===
A module ''P'' is projective if and only if every [[short exact sequence]] of modules of the form

:<math>0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0</math>

is a [[split exact sequence]]. That is, for every surjective module homomorphism {{nowrap|''f'' : ''B'' ↠ ''P''}} there exists a '''section map''', that is, a module homomorphism {{nowrap|''h'' : ''P'' → ''B''}} such that ''fh'' = id<sub>''P''</sub>. In that case, {{nowrap|''h''(''P'')}} is a [[direct summand]] of ''B'', ''h'' is an [[isomorphism]] from ''P'' to {{nowrap|''h''(''P'')}}, and {{nowrap|''hf''}} is a [[projection (linear algebra)|projection]] on the summand {{nowrap|''h''(''P'')}}.  Equivalently,

:<math>B = \operatorname{Im}(h) \oplus \operatorname{Ker}(f) \ \ 
\text{ where } \operatorname{Ker}(f) \cong A\ \text{ and }
\operatorname{Im}(h) \cong P.</math>

=== Direct summands of free modules ===

A module ''P'' is projective if and only if there is another module ''Q'' such that the [[direct sum of modules|direct sum]] of ''P'' and ''Q'' is a free module.

=== Exactness ===

An ''R''-module ''P'' is projective if and only if the covariant [[functor]] {{nowrap|Hom(''P'', -): ''R''-'''Mod''' → '''Ab'''}} is an [[exact functor]], where {{nowrap|''R''-'''Mod'''}} is the category of left ''R''-modules and '''Ab''' is the [[category of abelian groups]]. When the ring ''R'' is [[commutative ring|commutative]], '''Ab''' is advantageously replaced by {{nowrap|''R''-'''Mod'''}} in the preceding characterization. This functor is always [[left exact functor|left exact]], but, when ''P'' is projective, it is also right exact. This means that ''P'' is projective if and only if this functor preserves [[epimorphism]]s (surjective homomorphisms), or if it preserves finite [[colimit]]s.

===Dual basis===
A module ''P'' is projective if and only if there exists a set <math>\{a_i \in P \mid i \in I\}</math> and a set <math>\{f_i\in \mathrm{Hom}(P,R) \mid i\in I\}</math> such that for every ''x'' in ''P'', ''f''<sub>''i''</sub>(''x'') is only nonzero for finitely many ''i'', and <math>x=\sum f_i(x)a_i</math>.