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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.
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 we can think of P as a kind of projection in F: the module endomorphism in F that is the identity on P and 0 on Q is an idempotent matrix.
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.
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.)
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 iff for every surjective module homomorphism f : M → P there exists a module homomorphism h : P → M such that fh = idP. 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.
A basic motivation of the theory is that projective modules (at least over certain commutative rings) are analogues of vector bundleIn 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 topolos. This can be made precise for the ring of continuous real-valued functions on a compactSeveral specialized usages of the terms compact and compactness exist. Multiple definitions of the term "compact" are found in mathematics: The most common usage relates to topology, where one considers compact spaces . This article also includes the clos Hausdorff spaceIn topology and related branches of mathematics, Hausdorff spaces and preregular spaces are kinds of topological spaces. The conditions imposed are the most significant separation axioms. Definitions Suppose that X is a topological space. X is a Hausdorff, as well as for the ring of smooth functions on a compact smooth manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefor (see Swan's theoremAbstract algebra Algebraic topology Differential geometry Differential topology Swan's theorem relates vector bundles to projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like ve).
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 ringIn abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S one wants to construct some ring R and ring homomorphism from R to R such that the image of S consists of units (invertible, 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 commutative ring is locally free if and only if it is projective.
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