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Table of Comparison
| Aspect | Free Module | Projective Module |
|---|---|---|
| Definition | Direct sum of copies of ring R | Direct summand of a free module |
| Structure | Has a basis over R | May not have a basis but splits exact sequences |
| Exactness | Always projective, hence exact in Hom functors | Projective, preserves exactness under Hom |
| Characterization | All elements expressed uniquely via basis | Summand of a free module, satisfies lifting property |
| Examples | R^n (n copies of ring R) | Direct summand of R^n, e.g. idempotent-generated modules |
| Importance | Fundamental building blocks in module theory | Generalizes free modules, crucial in homological algebra |
| Relation | Subset of projective modules | Includes all free modules |
Introduction to Free and Projective Modules
Free modules are direct sums of copies of a ring, providing a basis similar to vector spaces and allowing unique coordinate representations of elements. Projective modules generalize free modules by being direct summands of free modules, ensuring exactness properties that facilitate lifting and splitting in module theory. Understanding free and projective modules is fundamental in algebra, particularly in module classification and homological algebra contexts.
Fundamental Definitions
A free module over a ring is a module that has a basis, meaning it is isomorphic to a direct sum of copies of the ring, allowing every element to be uniquely expressed as a linear combination of basis elements with coefficients from the ring. A projective module is defined as a direct summand of a free module, which implies it inherits properties like lifting homomorphisms and satisfies the splitting property in exact sequences. The fundamental distinction lies in free modules having a global basis, whereas projective modules need only be locally projective or summands of free modules without requiring a global basis.
Key Differences Between Free and Projective Modules
Free modules are direct sums of copies of the base ring, possessing a basis that allows every element to be uniquely expressed as a linear combination. Projective modules are direct summands of free modules, characterized by the property that every surjective module homomorphism onto them splits. Unlike free modules, projective modules need not have a basis, but they retain lifting properties essential for splitting exact sequences in module theory.
Characteristic Properties of Free Modules
Free modules are characterized by having a basis, making every element uniquely expressible as a finite linear combination of basis elements with coefficients from the ring. This property ensures that free modules are projective since they satisfy the lifting property relative to module homomorphisms. Unlike general projective modules, free modules do not require a retract condition; their structure is fully determined by their free generating set.
Characteristic Properties of Projective Modules
Projective modules are characterized by the lifting property, allowing any module homomorphism defined on a quotient to be lifted to the module itself, which free modules naturally satisfy as they are direct sums of copies of the ring. Unlike free modules, projective modules need not have a basis but always appear as direct summands of free modules, ensuring their exactness properties in short exact sequences. The split exactness criterion and the ability to factor through surjective morphisms uniquely distinguish projective modules in module theory.
Criteria for a Module to be Free
A free module over a ring R is characterized by having a basis, meaning it is isomorphic to a direct sum of copies of R, which provides a straightforward criterion: the existence of a linearly independent generating set spanning the module. Projective modules, while direct summands of free modules, need not possess a basis, and thus lack this strict linear independence condition. The key criterion for a module to be free is that every element can be uniquely expressed as a finite R-linear combination of basis elements, ensuring the module is both generated and linearly independent over R.
Criteria for a Module to be Projective
A module \( P \) is projective if it satisfies the lifting property, meaning for every surjective module homomorphism \( f: M \to N \) and any module homomorphism \( g: P \to N \), there exists a homomorphism \( h: P \to M \) such that \( f \circ h = g \). Every projective module is a direct summand of a free module, so a module is projective if and only if it can be expressed as a direct summand of some free module. The characterization of projective modules via exact sequences states that \( P \) is projective if the functor \(\mathrm{Hom}(P, -)\) is exact, or equivalently, every short exact sequence \(0 \to K \to L \to P \to 0\) splits.
Examples Illustrating Free and Projective Modules
Free modules over a ring R are direct sums of copies of R, exemplified by R^n, where each element has a unique coordinate representation. Projective modules include free modules as a subset but also contain modules like direct summands of free modules, such as a module P satisfying P Q R^n for some Q. A classical example illustrating projective but non-free modules involves idempotent-generated modules over non-local rings, where P is projective but lacks a basis, contrasting with free modules which always have a basis.
Applications in Algebra and Beyond
Free modules serve as fundamental building blocks in module theory, providing a basis for constructing more complex modules through direct sums, and their straightforward structure facilitates explicit computations in homological algebra. Projective modules generalize free modules by allowing direct summands of free modules, playing a crucial role in solving extension problems and lifting properties in ring theory and category theory. Applications extend beyond pure algebra, impacting algebraic topology through vector bundles classification and influencing computational algebra systems that rely on projective resolutions for algorithmic module decomposition.
Summary: Choosing Between Free and Projective Modules
Free modules have a basis, making them direct sums of copies of the ring and easy to work with in computations. Projective modules generalize free modules by allowing direct summands of free modules, providing more flexibility in module decomposition. Selecting between free and projective modules depends on whether explicit bases are needed or more general splitting properties are desired for exact sequences in module theory.
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