Intuitive Understanding of Projective Modules

Question

I was wondering if anyone could give me any sort of intuitive explanation of what a projective module is or a useful way to think about them. I know the definition(s) in terms of lifting, split exact sequence, direct summand but I have no intuitive understanding of what it means for a module to be projective?

Thanks for any help

Answer 1

A finitely generated projective $A$ module $M$ can be thought as a locally free module, in the sense that each $M_{\mathfrak p}$ is free for a prime ideal $\mathfrak{p}$ (or that it´s free at each point in $Spec(A)$). This intuition come from Serre-Swan theorem, which states roughly that the sections functor is an equivalence between projective modules and complex vector bundles (furthermore vector bundles are locally trivial, and trivial bundles correspond to free modules). Note that a finitely generated projective module such that $M_{\mathfrak p}$ is free and have the same rank at each localization need not to be free (you cannot glue everything as in sheaf to get a free thing).

I think that it´s useful to cite too that finitely projective modules over local rings are free (this fact is know as Kaplansky theorem) and that over Von Neumann regular rings you can glue everything to get a free module.

Answer 2

There are quite few-equivalent-definitions of projective module. I would like to think about projective modules in terms of the $\operatorname{Hom}$ functor and short exact sequences as both structures are fundamental in homological algebra (with the $\otimes$ functor).

A module $P$ over a ring $R$ is projective if the $\operatorname{Hom}(P,)_R$ functor is exact, i.e. sends short exact sequences to short exact sequences.

You can check that this means that $\operatorname{Hom}(P,)_R$ preserves surjective homomorphisms of $R$-modules, whence the name "projective" (this is probably more evident with the definition involving the lifting property).

I personally like the definition involving the splitting of short exact sequences, due to its beautiful simplicity. In fact, in homological algebra (considering graded structures, differentials etc...) being "split" becomes a non trivial property of a short exact sequence.