Projective dimension

In mathematics, particularly in abstract algebra and homological 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.

A free module is a projective module, but the converse may be wrong over some rings, such as Dedekind rings. However, every projective module is a free module over a principal ideal domain, and over a polynomial ring over a field or the integers (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 in line with category theory is the property of lifting that carries over from free to projective modules. We can summarize this lifting property as follows: 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 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.)

The 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.

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