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.

Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.

Definitions

Direct summands of free modules

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 P is the image of a projection of F; the module endomorphism in F that is the identity on P and 0 on Q is idempotent and projects F to P.

Lifting property

Another definition 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 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.

For modules, the lifting property can equivalently be expressed as follows: the module P is projective if and only if for every surjective module homomorphism f : M ↠ P there exists a module homomorphism h : P → M such that fh = id_P. 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. More explicitly, M = im(h) ⊕ ker(f), and im(h) is isomorphic to P.

Exactness

Perhaps the most insightful and certainly the most abstract characterisation of a projective R-module M is that it has the property that the functor Hom(M,-): R-Mod→ Set is (right) exact. (Note that it is always left exact.) Equivalently, we can demand that this functor preserves epi's (surjective homomorphisms) or that it preserves finite colimits. (The equivalence of this last criterion with the lifting property should be clear.)

Vector bundles and locally free modules

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 smooth manifold (see Swan's theorem).

Vector bundles are locally free. If there is some notion of "localization" which can be carried over to modules, such as is given at localization of a ring, one can define locally free modules, and the projective modules then typically coincide with the locally free ones. Specifically, a finitely generated module over a 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 which are locally free and not projective. For instance, a Boolean ring has all of its localizations isomorphic to F₂, the field of two elements, so any module over a Boolean ring is locally free, but there are some non-projective modules over Boolean rings. 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's 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.) However, it is true that over any commutative ring, R, a finitely presented module is projective if and only if it is locally free if and only if it is a flat module.^[1]

Properties

• Direct sums and direct summands of projective modules are projective.

• If e = e² is an idempotent in the ring R, then Re is a projective left module over R.

• 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.

• The category of finitely generated projective modules over a ring is an exact category. (See also algebraic K-theory).

• Every module over a field or skew field is projective (even free). A ring over which every module is projective is called semisimple.

• An abelian group (i.e. a module over Z) is projective if and only if it is a free abelian group. The same is true for all principal ideal domains; the reason is that for these rings, any submodule of a free module is free.

• Over a Dedekind domain a non-principal ideal is always a projective module that is not a free module.

• Over a direct product of rings R × S where R and S are nonzero rings, both R × 0 and 0 × S are non-free projective modules.

• Over a matrix ring M_n(R), the natural module Rⁿ is projective but not free. More generally, over any artinian semisimple ring, every module is projective, but the only free proper submodule of the regular module is the zero module.

• Every projective module is flat.^[2] The converse is in general not true: the abelian group Q is a Z-module which is flat, but not projective. ^[3]
• 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 difficult.

Serre's problem

The Quillen–Suslin theorem is another deep result; it states that if K is a field, or more generally a principal ideal domain, and R = K[X₁,...,X_n] 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 and Quillen and Suslin independently and simultaneously treated the case of finitely generated modules. Since every projective module over a principal ideal domain is free, it is attractive to think the following is true: if R is a commutative ring such that every (finitely generated) projective R-module is free then every (finitely generated) projective R[X]-module is free. This is false. A counterexample occurs with R equal to the local ring of the curve y² = x³ at the origin. So you cannot prove Serre's problem by a simple induction on the number of variables.

Projective resolutions

Given a module, M, a projective resolution of M is an infinite exact sequence of modules

· · · → P_n → · · · → P₂ → P₁ → P₀ → M → 0,

with all the P_i's projective. Every module possesses a projective resolution. In fact a free resolution (resolution by free modules) exists. Such an exact sequence may sometimes be seen written as an exact sequence P(M) → M → 0. The minimal length of a finite projective resolution of a module M is called its projective dimension and denoted pd(M). If M does not admit a finite projective resolution then the projective dimension is infinite. A classic example of a projective resolution is given by the Koszul complex K_•(x).

Notes

1. ^ Eisenbud D.:Commutative Algebra with a view towards Algebraic Geometry, corollary 6.6, GTM 150, Springer-Verlag, 1995.
2. ^ Hazewinkel, et. al. (2004), Corollary 5.4.5, p. 131.
3. ^ Hazewinkel, et. al. (2004), Remark after Corollary 5.4.5, p. 131–132.

References

• Iain T. Adamson (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. ISBN 0-05-002192-3. 
• Serge Lang (1993). Algebra (3rd ed. ed.). Addison–Wesley. ISBN 0-201-55540-9. 
• Hazewinkel, Michiel; Gubareni, Nadezhda Mikhaĭlovna; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004). Algebras, rings and modules. Springer. ISBN 9781402026904. .