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.
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
- Choose 0 → K → P → M → 0 with P projective.
- In the long exact sequence, LnT(P) = 0 for n ≥ 1.
- Exactness then gives LnT(M) ≅ Ln−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
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.
Vanishing on the ends
If the outer terms vanish, the middle map is an isomorphism. The commonest way to compute an unknown derived functor.
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.
Splicing
A long exact sequence can be cut into short exact sequences, each analysed separately, then reassembled.
Induction on dimension
If the module has finite projective dimension, the sequence terminates and induction on that dimension closes the argument.
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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