The same construction, a different notion of projective
Given a projective class, resolve by relative projectives, apply the functor, take homology. The result is a family of relative derived functors with all the expected properties: agreement in degree 0, long exact sequences for allowable short exact sequences, vanishing on relative projectives. What changes is the answer, because fewer objects are projective and more resolutions are needed. Hochschild cohomology is the standard instance.
Learning objectives
- Construct relative derived functors from a projective class.
- State which properties survive and which do not.
- Compute relative Ext in a simple case.
- Relate relative and absolute derived functors.
Section 01Construction and properties
- Fix a projective class (P, E) with enough projectives.
- For an object M, build a resolution by objects of P using allowable epimorphisms at every stage.
- Apply the additive functor T and take homology. Independence of the resolution follows from the relative comparison theorem.
- Long exact sequences arise for E-exact short exact sequences only, not for arbitrary ones.
- Degree 0 recovers T when T is right exact with respect to the relative structure.
This is the price of the relative theory. A short exact sequence that is exact but not allowable gives no long exact sequence at all, so the usual computational habit of embedding a module in any convenient sequence must be checked against the class in force.
Section 02Hochschild cohomology
For a k-algebra Λ, take the class of epimorphisms split over k. The relative projectives are summands of Λ ⊗k V, and the relative Ext of Λ-bimodules is Hochschild cohomology:
| Degree | Meaning |
|---|---|
| 0 | The centre of Λ, or the invariants of M |
| 1 | Derivations modulo inner derivations |
| 2 | Infinitesimal deformations of the algebra structure |
| 3 | Obstructions to extending a deformation |
HH² classifies first-order deformations and HH³ holds the obstructions. The same pattern as group and Lie algebra extensions, in the relative setting — and the reason Hochschild cohomology is central to noncommutative geometry and quantisation.
Section 03Comparing relative and absolute
A relative resolution is in particular a complex, and comparing it with an absolute one produces a spectral sequence relating the two families. When the relative projectives happen to be absolutely projective, the two theories coincide.
For a separable algebra, every epimorphism splits over k automatically, so Hochschild cohomology vanishes above degree 0 and matches the absolute answer.
Relative Ext is computed from fewer projectives, so it is generally larger. The difference is exactly what the comparison spectral sequence measures.
Maschke's theorem makes a group algebra semisimple, killing absolute cohomology; separability makes an algebra relatively projective over its base, killing Hochschild cohomology. The two conditions play the same structural role in their respective theories.
ReferenceFrequently asked questions
Why does relative Ext tend to be larger?
Because fewer objects count as projective, so resolutions are longer and more homology survives. In the extreme case where only split epimorphisms are allowable, every object is projective and the theory collapses to nothing — the opposite extreme.
Is Hochschild cohomology a special case of group cohomology?
No, and confusing them is a common error. For a group algebra k[G], Hochschild cohomology with coefficients in k[G] itself differs from group cohomology; the two agree only for particular coefficient bimodules, notably when the bimodule structure is twisted to be one-sided.
What is a cotriple homology theory?
A relative theory whose projective class comes from an adjunction: the cotriple generated by the adjunction produces canonical resolutions. Most algebraic homology theories — of commutative algebras, of monoids, of categories — arise this way.
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