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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