Projective module
Definition and Properties
A projective module is a type of module that arises in various areas of abstract algebra. Projective modules are characterized by a property that allows them to "lift" homomorphisms over epimorphisms. This lifting property makes projective modules particularly useful in the study of homological algebra, where they play a central role in the definition of projective resolutions and the computation of derived functors.
In more concrete terms, a projective module can be thought of as a generalization of a free module. Just as a free module is a module that can be generated by a set of free generators, a projective module is a module that can be obtained as a direct summand of a free module. This means that if P is a projective R-module, there exists a free R-module F and another R-module M such that F is isomorphic to the direct sum of P and M. This property is sometimes referred to as the "direct summand property" of projective modules.
Examples
One of the most basic examples of a projective module is the module R itself, where R is a ring. This is because R is a free module over itself, and hence a direct summand of a free module. Another example is the module R^n, the direct sum of n copies of R. This module is free, and hence projective, for any positive integer n.
A more complex example of a projective module is the module of all polynomials in one variable over a field. This module is projective over the ring of polynomials, because it can be obtained as a direct summand of the free module of all polynomials in two variables.
Characterizations
There are several equivalent characterizations of projective modules, each of which provides a different perspective on their structure and properties. These characterizations include:
1. A module P is projective if and only if it is a direct summand of a free module. 2. A module P is projective if and only if every short exact sequence of modules 0 → N → M → P → 0 splits. 3. A module P is projective if and only if for every epimorphism f: M → N and every homomorphism g: P → N, there exists a homomorphism h: P → M such that f ∘ h = g.
Each of these characterizations provides a different way of understanding the nature of projective modules. The first characterization emphasizes the connection between projective modules and free modules, the second characterization highlights the role of projective modules in the theory of exact sequences, and the third characterization underscores the lifting property that is central to the definition of projective modules.
Applications
Projective modules play a crucial role in many areas of abstract algebra, including homological algebra, category theory, and the theory of rings and modules. In homological algebra, projective modules are used to construct projective resolutions, which are tools for computing derived functors and Ext functors. In category theory, projective modules are important because they are examples of projective objects, which are objects that have the right lifting property with respect to epimorphisms. In the theory of rings and modules, projective modules are used to study the structure of modules and to investigate properties of rings.