Projective Classes of Epimorphisms

Executive summary

Absolute exactness is a choice, not a law

Ordinary homological algebra treats every epimorphism as available for resolutions. Relative homological algebra fixes a smaller class — those surjections that split after restriction to a subring, say — and declares only those allowable. The objects that lift along that class are the relative projectives, and the whole apparatus of resolutions and derived functors runs again with respect to it. The point is not generality for its own sake: several natural invariants are relative derived functors and are invisible to the absolute theory.

Learning objectives

Define a projective class and its associated allowable epimorphisms.
Identify relative projectives in the standard examples.
State the conditions making relative resolutions possible.
Explain why the absolute theory is a special case.

Section 01The definition

A projective class is a pair: a class P of objects and a class E of epimorphisms, each determining the other. P lies in P exactly when every map from P lifts along every member of E; an epimorphism lies in E exactly when every map from every member of P lifts along it. The pair must also have enough projectives: every object receives a member of E from an object of P.

Standard projective classes
Class of epimorphisms	Relative projectives	Theory obtained
All epimorphisms	Ordinary projectives	Absolute homological algebra
Split epimorphisms	All objects	Trivial theory — all derived functors vanish above degree 0
Epimorphisms split over a subring Γ ⊆ Λ	Direct summands of induced modules	Relative (Λ, Γ)-homological algebra
Epimorphisms split as abelian groups	Direct summands of Λ ⊗ A	Hochschild theory for an algebra
Pure epimorphisms	Pure-projectives	Pure homological algebra

The two extremes bracket everything

Taking all epimorphisms recovers the usual theory; taking only split ones makes every object projective and kills all higher derived functors. Every relative theory sits between these, and the interesting cases are the ones with a genuine subring or a genuine purity condition.

Section 02Relative exactness

A sequence is E-exact when it is exact and the relevant epimorphisms are allowable. Relative resolutions are built from members of P using allowable epimorphisms, and the comparison theorem holds verbatim with ‘projective’ and ‘surjection’ replaced by their relative versions.

Stage 01Fix the classChoose which epimorphisms are allowable, typically by a splitting condition after restriction.
Stage 02Identify relative projectivesUsually the direct summands of induced or extended objects — the ones for which lifting is formal.
Stage 03Build resolutionsEnough relative projectives means every object has a relative resolution.
Stage 04DeriveApply a functor, take homology. The comparison theorem gives independence of the chosen resolution.

Everything transfers because the proofs never used more

The comparison theorem, the horseshoe lemma and the long exact sequences use only the lifting property and the existence of enough projectives. Both hold by hypothesis in a projective class, so the entire machinery of the derived-functors stream applies unchanged.

Section 03Why it is needed

Motivation

Hochschild cohomology

For an algebra over a commutative ring k, the relevant class is epimorphisms split over k. Hochschild cohomology is the relative Ext, and it differs from the absolute Ext except in the separable case.

Motivation

Relative group cohomology

For H ≤ G, the class of epimorphisms split over ℤ[H] gives cohomology relative to H, which vanishes when H = G and measures how far G is from H.

Motivation

Amitsur and cotriple homology

Many homology theories — of algebras, of monoids, of categories — are relative derived functors for a class determined by an adjunction.

Relative and absolute answers genuinely differ

Hochschild cohomology of a group algebra is not group cohomology; relative Ext over a subring is not Ext over the ring. Quoting a result from one theory in the other is a real error, not a change of viewpoint, and the notation must always record which class is in force.

ReferenceFrequently asked questions

Is a projective class the same as an exact structure?

They are closely related. An exact category fixes a class of admissible short exact sequences; a projective class fixes the epimorphisms and enough projectives for them. Most relative theories can be phrased either way, and the exact-category language is the more common modern choice.

Does every class of epimorphisms give a projective class?

No — the class must be closed under the operations the lifting correspondence requires, and enough projectives must exist. Classes defined by a splitting condition after an exact restriction functor are the reliable source of examples.

Can the relative theory be computed from the absolute one?

Sometimes, via a spectral sequence comparing the two. The change-of-rings spectral sequences are of exactly this type, relating Ext over a ring to Ext over a subring.

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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 Projective Classes of Epimorphisms. 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 Projective Classes of Epimorphisms as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—relative, projective, exactness, section, classes—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 Projective Classes of Epimorphisms?