Projective module

In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). Such modules can be characterised in various ways, in some cases only for rings R that are commutative, and of special types (see Swan's theorem).

The easiest characterisation is as a direct summand of a free module. That is, a module P is projective provided there is a module Q such that the direct sum of the two is a free module F. From this it follows that we can think of P as a kind of projection in F: the module endomorphism in F that is the identity on P and 0 on Q is an idempotent matrix.

Another way that is more in line with category theory is to extract the property, of lifting, that carries over from free to projective modules. Using a basis of a free module F, it is easy to see that if we are given a surjective module homomorphism from N to M, the corresponding mapping from Hom(F,N) to Hom(F,M) is also surjective (it's from a product of copies of N to the product with the same index set for M). Using the homomorphisms P->F and F->P for a projective module, it is easy to see that P has the same property; and also that if we can lift the identity P->P to P->F for F some free module mapping onto P, that P is a direct summand. This gives an abstract characterisation, therefore.

Another way to pick out projective modules is as locally free; this assumes some idea of localisation, such as is given at localization of a ring, carried over to modules, to make sense. A basic motivation of the theory is that such modules are analogues of vector bundles - something that for rings C(X) for a compact Hausdorff space can be justified in detail.