Mathematics•Derived Functors
The Long Exact Homology Sequence
The connecting homomorphism, where it comes from, and why its naturality is the whole point.
One short exact sequence of complexes, one infinite exact sequence of homology
A short exact sequence of chain complexes does not give a short exact sequence of homology groups; it gives a long exact sequence, with a connecting map lowering degree by one. The connecting map is built by the snake lemma — lift a cycle, apply the differential, pull back — and the essential theorem is that it is well defined and natural. Every computational device in the subject is an application of this one sequence.
Learning objectives
- Construct the connecting homomorphism explicitly.
- Prove that it is well defined.
- State exactness of the resulting long sequence.
- Explain why naturality permits comparison arguments.
Section 01The construction
- Start with 0 → A• → B• → C• → 0, exact in each degree.
- Take a cycle c ∈ Cn and lift it to b ∈ Bn. Possible by surjectivity in degree n.
- Apply the differential: ∂b maps to ∂c = 0 in Cn−1, so ∂b comes from a unique a ∈ An−1.
- a is a cycle, because its image ∂∂b in B is zero and A → B is injective.
- Define ∂*[c] = [a]. Independence of the lift b is the content of the construction.
- The resulting sequence … → Hn(A) → Hn(B) → Hn(C) → Hn−1(A) → … is exact.
The connecting map lowers degree by one because it applies the differential once. In the cohomological convention it raises degree by one for the same reason. Every connecting homomorphism in the subject — in Ext, in Tor, in group cohomology — is this construction.
Section 02Naturality
Given a morphism of short exact sequences of complexes, the induced maps on homology commute with the connecting homomorphisms. So a morphism of short exact sequences yields a morphism of long exact sequences.
Existence of the long exact sequence computes one object. Naturality lets two computations be compared, which is what powers every induction and every five-lemma argument. When a source stresses that a sequence is natural, it is signalling that these arguments are available.
Section 03Applications
Mayer–Vietoris
In topology, a space covered by two open sets gives a short exact sequence of singular chain complexes, and the long exact sequence computes the homology of the union.
The pair sequence
A subspace inclusion gives a short exact sequence of chains and hence the long exact sequence of the pair, with relative homology in the middle.
Ext and Tor sequences
A short exact sequence of modules, resolved compatibly by the horseshoe lemma, produces a short exact sequence of complexes and hence the long exact sequences of derived functors.
Dimension shifting
Splicing a module into a short exact sequence with a projective term reduces degree n to degree n−1.
Five-term sequence
Truncating a spectral sequence gives a five-term exact sequence — the inflation–restriction sequence in group cohomology.
Bockstein
The connecting map of 0 → ℤ/p → ℤ/p² → ℤ/p → 0 is the Bockstein operator, detecting torsion of higher order.
ReferenceFrequently asked questions
Why is the sequence long rather than short?
Because homology is neither left nor right exact, so information leaks between degrees. The connecting map is exactly the leak, and tracking it in every degree produces an infinite sequence.
Does the long exact sequence terminate?
It continues in both directions in general. It terminates when the complexes are bounded, or when the relevant homology vanishes beyond some degree — for example over a ring of finite global dimension.
Is the connecting homomorphism canonical?
Yes, up to the sign convention. Different texts choose different signs, and comparisons across sources should check the convention before concluding two sequences disagree.
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Curated next steps from this page. The site also surfaces algorithmically related reading below.
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 The Long Exact Homology Sequence. 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 The Long Exact Homology Sequence as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—sequence, exact, long, short, complexes—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 The Long Exact Homology Sequence?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which assumption about sequence would change the result most if it proved false?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What evidence would allow an independent reviewer to reproduce or challenge the conclusion?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which boundary, exception or failure case has not yet been tested?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What must be handed over, monitored or reviewed after the immediate work is complete?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Authoritative references and use notes
The sources below were selected as institutional or primary guidance for the broader practice. They support the handbook method; they do not imply that every statement or clause in a source applies to every project. Confirm the current edition, jurisdiction, contract and application before treating any requirement as mandatory.
- MIT OpenCourseWare — Number Theory I — Massachusetts Institute of Technology. Used for algebraic and analytic number theory. Accessed 2026-08-13.
- MIT OpenCourseWare — Algebra I — Massachusetts Institute of Technology. Used for groups, vector spaces, linear transformations and linear groups. Accessed 2026-08-13.
- Page ID
- KV-MATH-0126
- Taxonomy
- ENG-MATH — Engineering / Mathematics
- Collection
- COL-HOMALG-001
- Topic stream
- HA-DERIVED
- Version
- 1.1.0 / content 2026.08
- Last reviewed
- 2026-08-06