Product plus torsion correction
Over a principal ideal domain, the homology of a tensor product of complexes is not simply the tensor product of the homologies. There is an additional Tor term, shifted down one degree, and the three fit into a short exact sequence that splits — but not naturally. The formula is the reason homology with field coefficients is multiplicative while integral homology is not, and the Tor term is exactly the interaction between torsion in the two factors.
Learning objectives
- State the Künneth short exact sequence.
- Identify the hypotheses needed on the complexes and the ring.
- Explain why the splitting is not natural.
- Specialise to field coefficients and to torsion-free homology.
Section 01The theorem
Let C and D be complexes over a PID with C flat — for example a complex of free modules. Then for each n there is a short exact sequence
The sequence is natural in both complexes and splits, but the splitting is not natural.
The homology of the tensor product is isomorphic to the direct sum of the two outer terms, but no such isomorphism can be chosen compatibly with maps of complexes. Any argument requiring functoriality must use the short exact sequence itself, not the splitting.
Section 02Hypotheses
| Hypothesis | Role | If dropped |
|---|---|---|
| Base ring is a PID | Tor vanishes above degree 1, so one correction term suffices | Over a general ring, a spectral sequence with infinitely many possible terms replaces the sequence |
| C is a complex of flat modules | Ensures the tensor product behaves correctly on cycles and boundaries | Free complexes are the usual case; without flatness the sequence fails |
| Boundaries are flat | In the usual proof, Bn(C) flat is what is really used | Over a PID submodules of free are free, so this is automatic |
| Complexes bounded below | Convergence and finiteness of the sums | Unbounded cases need care with totalisation |
Global dimension 1 means Torn = 0 for n ≥ 2, so exactly one correction term appears. Over a ring of higher global dimension the correction becomes a spectral sequence, which is the general Künneth spectral sequence.
Section 03Special cases
Field coefficients
Over a field every module is free, so Tor vanishes and homology is exactly multiplicative: Hn(C ⊗ D) = ⊕ Hp ⊗ Hq.
Torsion-free homology
If the homology of either complex is torsion-free, Tor1 vanishes and the same clean formula holds over ℤ.
Topological product
For CW complexes, singular chains of a product are chain equivalent to the tensor product, so Künneth computes H*(X × Y) from the factors.
Working over a field removes the Tor term entirely and makes homology a functor to graded vector spaces, where dimension counting is available. The price is losing torsion information, which is precisely what the integral computation retains.
ReferenceFrequently asked questions
Why is the Tor term shifted down one degree?
Because it arises as the connecting homomorphism in the long exact sequence relating cycles, boundaries and homology, and connecting maps shift degree by one. The shift is structural, not a normalisation choice.
Does Künneth hold for cohomology?
There is a dual statement with Ext replacing Tor and appropriate finiteness hypotheses, but it is less clean: cohomology does not commute with infinite products as readily, so finite generation is usually assumed.
What replaces the formula over a general ring?
A Künneth spectral sequence with E2 page given by Tor in all degrees, converging to the homology of the tensor product. It reduces to the short exact sequence exactly when global dimension is at most 1.
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