Integral homology determines everything, up to an extension
The universal coefficient theorems express homology and cohomology with coefficients in a module A in terms of integral homology. For homology the correction is a Tor term one degree down; for cohomology it is an Ext term one degree up. Both sequences split non-naturally. Together they say that integral homology determines all other coefficient versions — but only up to an extension, and the extension data is genuinely lost if the splitting is used carelessly.
Learning objectives
- State both universal coefficient theorems.
- Explain the degree shift in each direction.
- Identify when the correction term vanishes.
- Explain the consequence for cohomology detecting torsion.
Section 01The two theorems
For a chain complex of free modules over a PID:
| Homology | Cohomology | |
|---|---|---|
| Main term | Hn ⊗ A | Hom(Hn, A) |
| Correction | Tor1(Hn−1, A) | Ext1(Hn−1, A) |
| Correction position | On the right | On the left |
| Degree of correction | One lower | One lower, contributing to degree n |
| Splits | Yes, non-naturally | Yes, non-naturally |
In homology the correction is a quotient; in cohomology it is a submodule. Getting this backwards produces wrong conclusions about which classes are detected, and the error is not visible from the shape of the sequence alone.
Section 02When corrections vanish
Field coefficients
Both Tor and Ext vanish over a field, so homology and cohomology are dual vector spaces and everything is clean.
Free homology
If Hn−1 is free then Tor and Ext against it vanish, and both sequences reduce to their main terms.
Torsion homology, integral coefficients
Ext1(ℤ/m, ℤ) = ℤ/m, so torsion in degree n−1 appears in cohomology in degree n. This is why integral cohomology sees torsion one degree up.
Torsion in Hn−1 contributes to Hn, not to Hn−1. Reading off torsion from cohomology therefore requires the shift to be applied, and forgetting it is a standard source of errors when comparing homology and cohomology tables.
Section 03Why splitting is not natural
A splitting exists for each complex separately, but a map of complexes need not commute with chosen splittings. Concretely, in topology the induced maps on cohomology do not respect the decomposition into Hom and Ext parts.
Any statement about induced maps, cup products or naturality must be phrased using the short exact sequence. The direct sum decomposition is a statement about isolated objects and cannot be propagated along morphisms.
ReferenceFrequently asked questions
Why does cohomology have an Ext term and homology a Tor term?
Because cohomology applies Hom, which is left exact with Ext as its derived functor, while homology applies tensor, which is right exact with Tor as its derived functor. The position of the correction follows the exactness direction of the functor involved.
Can cohomology be recovered from homology completely?
Up to the extension recorded by the short exact sequence. With field coefficients the answer is yes and cohomology is the dual of homology; integrally, the extension can be non-trivial and the two carry genuinely different information.
What is the dual Künneth formula?
The statement computing the cohomology of a product in terms of the cohomology of the factors, with an Ext correction term, under finiteness hypotheses. It is less symmetric than the homology version because Hom does not commute with infinite direct sums.
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