Relative Derived Functors

Executive summary

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

AlgorithmRelative derived functorsin: a projective class and a functor → out: relative derived functors

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.

Long exact sequences are restricted

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:

HHⁿ(Λ, M) = Extⁿ_{Λ^e, k}(Λ, M)

Interpretation in low degrees
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

Deformation theory lives here

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.

They agreeWhen relative projectives are projective

For a separable algebra, every epimorphism splits over k automatically, so Hochschild cohomology vanishes above degree 0 and matches the absolute answer.

They differIn general

Relative Ext is computed from fewer projectives, so it is generally larger. The difference is exactly what the comparison spectral sequence measures.

Separability is the relative analogue of semisimplicity

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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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 Relative Derived Functors. 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 Relative Derived Functors as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—relative, derived, functors, construction, section—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 Relative Derived Functors?