Mathematics•Extensions, Ext and Tor
The Two Long Exact Sequences of Ext
One for each variable, running in opposite directions — and why keeping them straight is the main practical skill.
Two sequences, opposite directions, same connecting construction
Ext is a bifunctor, so a short exact sequence in either argument yields a long exact sequence. In the second variable the sequence follows the original direction; in the first, contravariance reverses it, so the sub-object contributes on the right. Both are natural, which is what permits comparison arguments, and both begin with the corresponding Hom sequence — making the failure of Hom's exactness visible as the first connecting map.
Learning objectives
- Write both long exact sequences correctly.
- Explain the direction reversal in the first variable.
- Identify where the Hom sequence sits inside each.
- Use naturality to compare two long exact sequences.
Section 01The second variable
From 0 → A′ → A → A″ → 0 and a fixed C:
The direction matches the original sequence throughout, because the functor is covariant. The first connecting map measures exactly which homomorphisms into A″ fail to lift to A.
Section 02The first variable
From 0 → C′ → C → C″ → 0 and a fixed A:
The quotient C″ now appears first and the submodule C′ second — the reverse of the second-variable sequence. Writing the sequence in the wrong order is the most common error in Ext computations, and it produces conclusions that look plausible. Fix the variable in writing before starting.
| Second variable (covariant) | First variable (contravariant) | |
|---|---|---|
| Order of outer terms | A′, A, A″ — unchanged | C″, C, C′ — reversed |
| Connecting map raises degree | Yes | Yes |
| Interpretation of ∂ in degree 0 | A map to A″ that does not lift | A map out of C′ that does not extend |
| Vanishes when | A′ injective | C″ projective |
Section 03Naturality and comparison
- Take a morphism between two short exact sequences in the same variable.
- Both rows produce long exact Ext sequences.
- Naturality of the connecting homomorphism makes every square commute, giving a morphism of long exact sequences.
- Apply the five lemma at each position: if two out of three vertical maps are isomorphisms, so is the third.
- Conclude by induction along the sequence. This is the shape of most proofs in the subject.
Almost no Ext group is computed from a resolution in practice. The normal method is to embed the module in a short exact sequence with known terms and read the unknown group off the long exact sequence. Choosing that sequence well is the whole skill.
ReferenceFrequently asked questions
Do both sequences terminate?
They continue indefinitely to the right. They terminate in practice when the modules have finite projective or injective dimension — over a PID both stop after Ext1, which is why abelian group computations are short.
Can the two sequences be combined?
Not into a single long exact sequence, but a short exact sequence in each variable simultaneously produces a commutative diagram of long exact sequences, and the resulting spectral sequence is one route to the balance theorem.
What does the connecting map actually do?
In degree 0 of the second-variable sequence it takes a homomorphism C → A″ and returns the extension of C by A′ obtained by pulling back the given short exact sequence along it. The obstruction to lifting is realised as an explicit extension class.
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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 Two Long Exact Sequences of Ext. 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 Two Long Exact Sequences of Ext as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—long, exact, sequences, connecting, 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 The Two Long Exact Sequences of Ext?
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 long 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-0121
- Taxonomy
- ENG-MATH — Engineering / Mathematics
- Collection
- COL-HOMALG-001
- Topic stream
- HA-EXT-TOR
- Version
- 1.1.0 / content 2026.08
- Last reviewed
- 2026-08-06