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). Various equivalent characterizations of these modules are available.
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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.
We can summarize this lifting property as follows: a module P is projective if and only if for any surjective module homomorphism f : N → M and every module homomorphism g : P → M, there exists a homomorphism h : P → N such that fh = g. (We don't require the lifting homomorphism h to be unique; this is not a universal property.)
For modules, the lifting property can equivalently be expressed as follows: the module P is projective iff for every surjective module homomorphism f : M → P there exists a module homomorphism h : P → M such that fh = idP. The existence of such a section map h implies that P is a direct summand of M and that f is essentially a projection on the summand P.
A basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of vector bundles. This can be made precise for the ring of continuous real-valued functions on a compact Hausdorff space, as well as for the ring of smooth functions on a compact smooth manifold (see Swan's theorem).
Vector bundles are locally free. If there is some notion of "localization" for our rings, such as is given at localization of a ring, which can be carried over to modules, one can often define locally free in more general settings, and the projective modules are then just the locally free ones.
Direct sums and summands of projective modules are projective. Submodules of projective modules need not be projective; a ring R for which every submodule of a projective left module is projective is called left hereditary.
In line with the above intuition of "locally free = projective" is the following theorem due to Kaplansky: over a local ring R, every projective module is free. This is easy to prove for finitely generated projective modules, but the general case is deep.
The Quillen-Suslin theorem is another deep result; it states that if K is a field and R = K[X1,...,Xn] is a polynomial ring over K, then every projective module over R is free.Definitions
Direct summands of free modules
Lifting property
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N ------->>MThe advantage of this definition of "projective" is that it can be carried out in categories more general than module categories: we don't need a notion of "free object". It can also be dualized, leading to injective modules.Locally free
Facts
