The Long Exact Sequences of Derived Functors

Executive summary

The computational engine of the whole subject

A short exact sequence of modules is resolved compatibly by the horseshoe lemma, giving a degreewise split short exact sequence of complexes. Applying an additive functor preserves the splitting, so exactness survives, and the long exact homology sequence then produces the long exact sequence of derived functors. Almost every computation in homological algebra is an application of this, usually combined with a vanishing result and the five lemma.

Learning objectives

Derive the long exact sequence from the horseshoe lemma.
Apply dimension shifting to reduce degree.
Use vanishing on projectives to compute unknown terms.
Combine naturality with the five lemma.

Section 01Derivation

Stage 01HorseshoeResolve the outer terms, then build a compatible resolution of the middle. The result is degreewise split.
Stage 02Apply the functorDegreewise splitness survives any additive functor, so the sequence of complexes remains exact.
Stage 03Long exact homology sequenceApply the snake-lemma construction degree by degree.
Stage 04Read offThe result is the long exact sequence of derived functors, natural in the original short exact sequence.

… → L_nT(A) → L_nT(B) → L_nT(C) → L_n−1T(A) → … → T(C) → 0

Degreewise splitness, not plain exactness

An arbitrary additive functor need not preserve exactness — that is the whole problem being addressed. It does preserve split exactness, because splitting is an identity between morphisms. The horseshoe lemma is designed to deliver exactly this.

Section 02Dimension shifting

AlgorithmReducing degree by onein: L_nT(M) → out: L₁T of a syzygy

Choose 0 → K → P → M → 0 with P projective.
In the long exact sequence, L_nT(P) = 0 for n ≥ 1.
Exactness then gives L_nT(M) ≅ L_n−1T(K) for n ≥ 2. The isomorphism is the connecting homomorphism.
In degree 1, L₁T(M) is the kernel of T(K) → T(P).
Iterate to reduce any degree to degree 1 of a higher syzygy.

Section 03Standard patterns

Pattern

Two known, one unknown

If two of three consecutive terms are known, exactness determines the third up to an extension — and often exactly, if one neighbour vanishes.

Pattern

Vanishing on the ends

If the outer terms vanish, the middle map is an isomorphism. The commonest way to compute an unknown derived functor.

Pattern

Comparison by five lemma

A map of short exact sequences gives a map of long exact sequences; two isomorphisms out of three force the third.

Pattern

Splicing

A long exact sequence can be cut into short exact sequences, each analysed separately, then reassembled.

Pattern

Induction on dimension

If the module has finite projective dimension, the sequence terminates and induction on that dimension closes the argument.

Pattern

Reduction to cyclic modules

Additivity plus a filtration reduces many statements to modules of the form Λ/J.

ReferenceFrequently asked questions

What if the functor is contravariant?

The sequence reverses: the quotient contributes first and the submodule last, exactly as in the first-variable Ext sequence. The construction is identical; only the bookkeeping of which term goes where changes.

Does the sequence always terminate on the right?

For left derived functors of a right exact functor, yes — it ends with T(C) → 0. For right derived functors it begins with 0 → T(A) and continues indefinitely to the right.

Can I always compute the connecting homomorphism explicitly?

In principle yes, by following the snake lemma construction through the chosen resolutions. In practice it is usually characterised by naturality and by what it does on a generator, which is enough for most arguments.

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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 The Long Exact Sequences of 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 The Long Exact Sequences of Derived Functors as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—derived, dimension, long, exact, functors—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 Sequences of Derived Functors?