projective module

A projective module is a module that is almost free — free enough for the purposes that matter. The cleanest description: it is a direct summand of a free module, a piece that splits off cleanly. The name comes from a lifting property: maps out of a projective module can always be 'projected up' through any surjection, the way light through a slide can be focused back onto a chosen plane.

Three equivalent definitions for an R-module P: (i) P is a direct summand of a free module, i.e. there is Q with P ⊕ Q free; (ii) every surjection M -> P splits, so P is a direct summand of any module mapping onto it; (iii) the lifting property: for every surjection g : M -> N and every map f : P -> N, there is a lift h : P -> M with g∘h = f. Equivalently, the functor Hom(P, -) is exact.

Free implies projective, and over many rings the converse holds — over a PID, a local ring, or a polynomial ring over a field (the Quillen-Suslin theorem) every finitely generated projective is free. But not always: over Z/6Z the ideal 2Z/6Z is projective but not free, and over the ring of integers of a number field the nonprincipal ideals are projective and non-free, with their failure to be free measured by the class group.