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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, a free module is a module having a free basis. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg. Henri Cartan (born July 8, 1904) is a son of Élie Cartan, and is, as his father was, a distinguished and influential French mathematician. ... Samuel Eilenberg (September 30, 1913-January 30, 1998) was a Polish mathematician. ...

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. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, a projection is any one of several different types of functions, mappings, operations, or transformations, for example, the following: A set-theoretic operation typified by the jth projection map, written , that takes an element of the cartesian product to the value . ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...

Lifting property

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. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

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.) In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

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. Image File history File links Projective_module. ... In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. ...

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, P = im(h) ⊕ ker(f), and im(h) is isomorphic to M. ↔ ⇔ ≡ logical symbols representing iff. ...

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 compact smooth manifold (see Swan's theorem). In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... Swans theorem relates vector bundles to projective modules and gives rise to a common intuition throughout mathematics: projective modules over commutative rings are like vector bundles on compact spaces. Differential geometry Suppose M is a compact C∞-manifold, and a smooth vector bundle V is given on M. The...

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.) In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ... 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). ... In mathematics, a module is a finitely-generated module if it has a finite generating set. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists of idempotent elements. ...

Facts

• 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.

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

• 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.

In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In mathematics, a ring is called hereditary if all submodules of projective modules are again projective. ... In mathematics, algebraic K-theory is an advanced part of homological algebra concerned with defining and applying a sequence Kn(R) of functors from rings to abelian groups, for n = 0,1,2, ... . Here for traditional reasons the cases of K0 and K1 are thought of in somewhat different terms... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... The integers are commonly denoted by the above symbol. ... ↔ ⇔ ≡ logical symbols representing iff. ... In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In abstract algebra, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...

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 conjecture by a simple induction on the number of variables. The Quillen-Suslin theorem is a theorem in abstract algebra about the relationship between free modules and projective modules. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

Projective resolutions

Given a module, M, a projective resolution of M is an exact sequence (possibly infinite) of modules In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

· · · → 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). In mathematics, a free module is a module having a free basis. ... In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul. ...