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Concept
Projective module
Summary
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.
Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a polynomial ring (this is the Quillen–Suslin theorem).
Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.
The usual definition is in terms of the property of lifting that carries over from free to projective modules: a module P is projective if and only if for every surjective module homomorphism f : N ↠ M and every module homomorphism g : P → M, there exists a module homomorphism h : P → N such that f h = g. (We don't require the lifting homomorphism h to be unique; this is not a universal property.)
The advantage of this definition of "projective" is that it can be carried out in more general than module categories: we don't need a notion of "free object". It can also be , leading to injective modules. The lifting property may also be rephrased as every morphism from to factors through every epimorphism to . Thus, by definition, projective modules are precisely the projective objects in the .
A module P is projective if and only if every short exact sequence of modules of the form
is a split exact sequence. That is, for every surjective module homomorphism f : B ↠ P there exists a section map, that is, a module homomorphism h : P → B such that f h = idP . In that case, h(P) is a direct summand of B, h is an isomorphism from P to h(P), and h f is a projection on the summand h(P). Equivalently,
A module P is projective if and only if there is another module Q such that the direct sum of P and Q is a free module.
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