Projective module
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 (multivariate) polynomial ring over a field (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.An R-module P is projective if and only if the covariant functor Hom(P, -): R-Mod → Ab is an exact functor, where R-Mod is the category of left R-modules and Ab is the category of abelian groups.When the ring R is commutative, Ab is advantageously replaced by R-Mod in the preceding characterization.The left-to-right implications are true over any ring, although some authors define torsion-free modules only over a domain.The converse is true in the following cases: In general though, projective modules need not be free: The difference between free and projective modules is, in a sense, measured by the algebraic K-theory group K0(R); see below.[1] The converse is in general not true: the abelian group Q is a Z-module that is flat, but not projective.[2] Conversely, a finitely related flat module is projective.In general, the precise relation between flatness and projectivity was established by Raynaud & Gruson (1971) (see also Drinfeld (2006) and Braunling, Groechenig & Wolfson (2016)) who showed that a module M is projective if and only if it satisfies the following conditions: This characterization can be used to show that if[4] In other words, the property of being projective satisfies faithfully flat descent.The length of a finite resolution is the index n such that Pn is nonzero and Pi = 0 for i greater than n. If M admits a finite projective resolution, the minimal length among all finite projective resolutions of M is called its projective dimension and denoted pd(M).In this situation, the exactness of the sequence 0 → P0 → M → 0 indicates that the arrow in the center is an isomorphism, and hence M itself is projective.Projective modules over commutative rings have nice properties.The converse is true for finitely generated modules over Noetherian rings: a finitely generated module over a commutative Noetherian ring is locally free if and only if it is projective.However, there are examples of finitely generated modules over a non-Noetherian ring that are locally free and not projective.One example is R/I where R is a direct product of countably many copies of F2 and I is the direct sum of countably many copies of F2 inside of R. The R-module R/I is locally free since R is Boolean (and it is finitely generated as an R-module too, with a spanning set of size 1), but R/I is not projective because I is not a principal ideal.(If a quotient module R/I, for any commutative ring R and ideal I, is a projective R-module then I is principal.)[5] Moreover, if R is a Noetherian integral domain, then, by Nakayama's lemma, these conditions are equivalent to Let A be a commutative ring.If B is a (possibly non-commutative) A-algebra that is a finitely generated projective A-module containing A as a subring, then A is a direct factor of B.[7] Let P be a finitely generated projective module over a commutative ring R and X be the spectrum of R. The rank of P at a prime idealA basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of vector bundles.The Quillen–Suslin theorem, which solves Serre's problem, is another deep result: if K is a field, or more generally a principal ideal domain, and R = K[X1,...,Xn] is a polynomial ring over K, then every projective module over R is free.This problem was first raised by Serre with K a field (and the modules being finitely generated).Bass settled it for non-finitely generated modules,[8] and Quillen and Suslin independently and simultaneously treated the case of finitely generated modules.A counterexample occurs with R equal to the local ring of the curve y2 = x3 at the origin.Thus the Quillen–Suslin theorem could never be proved by a simple induction on the number of variables.