Kunneth formula for homology
For any and any module over a principal ideal domain for coefficients, we have:
Here, is torsion of modules over the ring .
- Kunneth formula for cohomology
- Universal coefficients theorem for homology
- Universal coefficients theorem for cohomology
- Dual universal coefficients theorem (computes cohomology in terms of homology)
Case of free modules
If all the homology groups are free (or more generally torsion-free) modules over , and/or all the homology groups , are free (or more generally torsion-free) modules over , then all the torsion part vanishes and we get:
In particular, if all and are free modules over and and denote the respective free ranks, and all these are finite, we obtain that:
Note that if is a field, then the above holds.
Impact for ranks even in case of torsion
When is a principal ideal domain and all the homologies are finitely generated modules over , we can consider the rank as the rank of the torsion-free part of the homology modules. If denotes the free rank of the torsion-free part of , we get:
Note that this applies even if the homology modules have torsion.
In the special case that , the numbers are called Betti numbers, and we get:
In particular, this yields that Poincare polynomial of product is product of Poincare polynomials.
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