Homology measures the failure of exactness, degree by degree
A chain complex is a sequence of modules with maps whose consecutive composites are zero, so boundaries sit inside cycles. The quotient is homology, and it vanishes in a degree exactly when the complex is exact there. Every construction that follows — resolutions, derived functors, spectral sequences — is an operation on complexes, and complexes themselves form an abelian category, which is what makes the theory self-applicable.
Learning objectives
- Define chain and cochain complexes and their differentials.
- Compute homology as cycles modulo boundaries.
- Explain the relation between exactness and vanishing homology.
- Describe why complexes form an abelian category.
Section 01Complexes
The condition ∂∂ = 0 says im ∂n+1 ⊆ ker ∂n, so the quotient makes sense:
| Term | Meaning |
|---|---|
| Cycle Zn | ker ∂n — killed by the differential |
| Boundary Bn | im ∂n+1 — hit by the differential |
| Homology Hn | Cycles modulo boundaries |
| Acyclic | All homology vanishes — the complex is exact |
| Cochain complex | Differential raises degree; homology is written Hn |
| Non-negative complex | Cn = 0 for n < 0 — the case for resolutions |
A complex is exact at degree n exactly when Hn = 0. Homology is therefore a graded measurement of non-exactness, and every long exact sequence in the subject is a statement that certain homology groups fit together.
Section 02Homology as a functor
A chain map f: C → D is a family commuting with the differentials. It carries cycles to cycles and boundaries to boundaries, so it induces Hn(f) on homology, and this assignment is functorial.
A short exact sequence of complexes does not give a short exact sequence of homology groups. It gives a long exact sequence, with a connecting homomorphism supplied by the snake lemma. This is not a defect to be repaired — the long exact sequence is the whole point.
Section 03The category of complexes
Complexes in an abelian category, with chain maps, form an abelian category: kernels, cokernels and exactness are all computed degreewise.
Homological algebra on complexes
Resolutions of complexes, hyperhomology and derived categories all become available, because the machinery applies to its own objects.
Degreewise exactness
A sequence of complexes is exact exactly when it is exact in each degree. This is what lets the snake lemma be applied degree by degree.
Double complexes
A complex of complexes is a double complex, and totalising it produces the Künneth and Grothendieck spectral sequences.
Inverting the chain maps that induce isomorphisms on homology — the quasi-isomorphisms — produces the derived category. That construction treats complexes, not modules, as the primary objects, and it is where modern homological algebra begins.
ReferenceFrequently asked questions
Why require the composite of consecutive differentials to be zero?
Because without it boundaries need not lie inside cycles and the quotient is undefined. The condition is exactly what makes homology a well-formed measurement, and it is satisfied by every construction that arises naturally — boundary maps in topology, differentials in resolutions.
What is the difference between chain and cochain complexes?
Only the direction of the differential and hence the indexing convention. A cochain complex is a chain complex with the indexing negated. Both appear because homology arises covariantly and cohomology contravariantly.
Can homology be non-zero in infinitely many degrees?
Yes. Bounded complexes are common in practice, but unbounded complexes with homology in every degree occur — for instance the cohomology of a group of infinite cohomological dimension.
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