Definition of Group Homology and Cohomology

Executive summary

Two functors, two families of derived functors

Invariants A^G is left exact; coinvariants A_G is right exact. Deriving them gives cohomology and homology respectively, and both are computed by resolving the trivial module ℤ over ℤ[G] — so Hⁿ(G, A) = Extⁿ(ℤ, A) and H_n(G, A) = Tor_n(ℤ, A). One resolution serves both.

Learning objectives

Define invariants and coinvariants and state their exactness.
Express group (co)homology as Ext and Tor over the group ring.
State the values in degree 0.
Write the long exact sequences in the coefficient variable.

Section 01The two functors

InvariantsA^G = { a : ga = a }

Left exact. Right derived functors give cohomology Hⁿ(G, A) = Extⁿ_ℤ[G](ℤ, A).

CoinvariantsA_G = A / (ga − a)

Right exact. Left derived functors give homology H_n(G, A) = Tor^ℤ[G]_n(ℤ, A).

Degree 0 and 1
Degree	Cohomology	Homology
0	A^G, the invariants	A_G, the coinvariants
1, trivial coefficients	Hom(G, A) — homomorphisms G → A	G^ab ⊗ A for A = ℤ: G^ab
2, trivial coefficients	Classifies central extensions	The Schur multiplier for A = ℤ

One resolution, two computations

Take a projective resolution of the trivial module ℤ over ℤ[G]. Applying Hom_ℤ[G](−, A) gives cohomology; applying − ⊗_ℤ[G] A gives homology. The same resolution serves both, which is why they are always developed together.

Section 02Long exact sequences

A short exact sequence of coefficient modules gives long exact sequences in both theories:

0 → A^G → B^G → C^G → H¹(G, A) → …

The first connecting map is the obstruction to lifting an invariant element of C to an invariant element of B — the prototypical use of H¹ as an obstruction group.

Where the theory earns its keep

Invariants of a quotient are not the quotient of the invariants. The failure is measured by H¹, and this single observation is what makes group cohomology useful in Galois theory, in arithmetic and in the descent arguments of algebraic geometry.

Section 03Basic properties

Property

Finite groups

|G| annihilates Hⁿ(G, A) for n ≥ 1. So cohomology of a finite group is torsion in positive degrees, and vanishes when |G| is invertible in A.

Property

Free groups

Hⁿ = 0 for n ≥ 2. Free groups have cohomological dimension 1, matching the fact that their classifying space is a graph.

Property

Trivial group

H⁰(1, A) = A and Hⁿ = 0 for n ≥ 1.

Property

Induced coefficients

Shapiro's lemma reduces cohomology with induced coefficients to a subgroup computation.

Property

Direct products

Künneth applies, since ℤ[G × H] = ℤ[G] ⊗ ℤ[H].

Property

Topological interpretation

H^*(G, ℤ) is the cohomology of the classifying space BG, an Eilenberg–MacLane space K(G, 1).

ReferenceFrequently asked questions

Why does |G| annihilate positive-degree cohomology?

Because the composite of restriction to the trivial subgroup and corestriction back is multiplication by |G|, and the middle term vanishes in positive degrees. The transfer argument is short and is the standard proof.

Is group cohomology the cohomology of a space?

Yes — of the classifying space BG, which is an Eilenberg–MacLane space K(G, 1). The algebraic and topological definitions agree, and each supplies techniques the other lacks.

Do homology and cohomology determine each other?

Via a universal coefficient theorem, up to an extension involving Ext. For finite groups there are additional duality relations, and Tate cohomology unifies the two into a single ℤ-graded theory.

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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 Definition of Group Homology and Cohomology. 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 Definition of Group Homology and Cohomology as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—group, cohomology, functors, homology, groups—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 Definition of Group Homology and Cohomology?