Projective module

Summary

A projective module is a module $P$ such that every linear map from $P$ lifts along every surjective linear map. Equivalently, $P$ is a direct summand of a free module. Projective modules keep the most useful property of free modules, lifting of maps, but need not have a basis. They are the building blocks of projective resolutions in homological algebra, and finitely generated projective modules correspond to vector bundles in geometry.