Projective objects and projective modules

(projective object)

An object $P$ in an abelian category is projective if the functor $\Hom(P, \wait)$ is exact. Equivalently, every map $P \to B$ lifts along any epimorphism $A \surjects B$. In $\mods{A}$ these are the projective modules, and free modules are projective.

A module is projective if and only if it is a direct summand of a free module. Over a local ring or a PID, projective modules are free; over $\ZZ/6$ the summand $\ZZ/2$ is projective and not free. $\mods{A}$ has enough projectives, since every module is a quotient of a free one; so the left derived functors of every right exact additive functor on $\mods{A}$, such as $\Tor$, are defined.

For $X=\PP^1_k$, neither $\mods{\OO_X}$ nor the category of quasicoherent $\OO_X$-modules has enough projectives [@Har10a, Exercise III.6.2], whereas $\mods{\OO_X}$ has enough injectives on every ringed space [@Har10a, Proposition III.2.2].

What is a projective object in an abelian category?

Which modules are projective, and how do projectives relate to free modules?

Does the category of $\OO_X$-modules have enough projectives?

Dependencies

D-DEFABCATAbelian categoriesuses

Backlinks

D-DEFDERIVDerived functorsuses