Free and Projective Modules

Executive summary

Every surjection onto a projective splits

A free module has a basis, so a homomorphism out of it is determined by arbitrary choices on the basis elements — which means any map into a quotient lifts. A projective module keeps the lifting property while giving up the basis, and it is exactly a direct summand of a free module. That is all resolutions need: every module is a quotient of a free one, so projective resolutions always exist, and Hom(P, −) is exact precisely when P is projective.

Learning objectives

Define free modules by their universal property.
State the lifting property defining projectivity.
Prove that projective is equivalent to being a direct summand of a free module.
Construct a projective resolution of an arbitrary module.
Give an example of a projective module that is not free.

Section 01Free modules

The free module on a set S is the direct sum of copies of Λ indexed by S. Its universal property is that a homomorphism out of it is exactly a function from S to the target:

Hom_Λ(F(S), N) ≅ Map(S, N)

Every module is a quotient of a free module — take S to be any generating set, or the underlying set of the module itself. This single fact is what guarantees that projective resolutions exist for every module over every ring.

Free means arbitrary choices are allowed

Defining a map out of a free module requires no compatibility checking whatsoever: pick images for the basis and extend. Every lifting argument in the subject ultimately reduces to this freedom.

Section 02Projectivity

P is projective when every diagram with a surjection B ↠ C and a map P → C admits a lift P → B.

AlgorithmEquivalent characterisations of projectivityin: a module P → out: whether P is projective

P is projective: every map out of P lifts along every surjection.
Every short exact sequence 0 → A → B → P → 0 splits. Apply the lifting property to the identity of P.
P is a direct summand of a free module. Take a free F with F ↠ P and split it.
Hom(P, −) is an exact functor.
P satisfies the dual basis lemma: there are elements and functionals reproducing each element as a finite sum.
All five conditions are equivalent.

Projective need not be free

Over ℤ/6ℤ, the ideal generated by 2 is a direct summand and hence projective, but it is not free — it has the wrong number of elements. Over a Dedekind domain, non-principal ideals are projective and non-free, and the class group measures precisely how many such modules exist.

Section 03Rings where the distinction collapses

When projective implies free
Ring	Projective ⇒ free?	Note
Field	Yes	Every module is free — a vector space
Principal ideal domain	Yes, for finitely generated	Submodules of free are free
Local ring	Yes	Kaplansky's theorem, even without finite generation
Polynomial ring over a field	Yes	Quillen–Suslin, resolving Serre's problem
Dedekind domain	No	Ideals are projective; class group counts the failure
ℤ/nℤ for composite n	No	Idempotents split off summands
Group ring ℤ[G]	Generally no	Relevant to modular representation theory

Section 04Projective resolutions

AlgorithmConstructing a projective resolutionin: a module M → out: a projective resolution of M

Choose a surjection ε: P₀ ↠ M with P₀ projective — a free module always works.
Set K₀ = ker ε, the first syzygy.
Choose a surjection P₁ ↠ K₀ and compose with the inclusion to get d₁: P₁ → P₀.
Repeat on ker d₁, and so on. The process never obstructs, because every module is a quotient of a free one.
The result … → P₂ → P₁ → P₀ → M → 0 is exact.

Why resolutions are the right idea

M itself may behave badly under a functor, but each P_n behaves perfectly. Applying the functor to the resolution rather than to M, and taking homology, produces invariants that agree with the functor in degree 0 and record its failure above.

ReferenceFrequently asked questions

Is every projective module a direct sum of finitely generated ones?

Over many rings yes, and Kaplansky proved that any projective module is a direct sum of countably generated ones. Over an arbitrary ring the structure can be complicated, and this is one reason finitely generated hypotheses appear so often.

Why not always use free resolutions?

They exist and are perfectly valid, and many texts use them exclusively. Projective resolutions are preferred because they can be much smaller, and because the projective property is what the proofs actually use — freeness is a convenience, not a necessity.

What is a syzygy?

The kernel at each stage of a resolution — the module of relations among the chosen generators. Iterating gives higher syzygies, and Hilbert's syzygy theorem bounds how long the process continues over a polynomial ring.

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Curated next steps from this page. The site also surfaces algorithmically related reading below.

ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Free and Projective Modules. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Free and Projective Modules as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—projective, modules, free, section, lifting—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Free and Projective Modules?