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