Künneth theorem

Named after Hermann Künneth set of Kiinneth is a set of the mathematical subfield of homological algebra. The set of attributes the homology of a tensor product of chain complexes on the homologies of the chain complexes involved in memorable formulation, he says that the homology of a tensor product of chain complexes up to torsion is equal to the tensor product of the homologies. The set of Kiinneth, which is often simply called the Künnethformel, is a generalization of the universal Koeffiziententheorems.

Tensor products of chain complexes

Are and two chain complexes, so the tensor product of chain complex is with

Specifically, is a chain complex, the only 0-th site has a module different from 0, then the chain complex

For this chain complex to write for short as well.

The vorzustellende here sentence answers the question of how to compute the homology of the tensor product of the homology of the chain complexes. In general, the homology of the tensor is not determined by the homology and, in addition, are essential ingredients to put on the ring and to the given chain complexes. The simplest formula for such a function would be that the -th homology of the tensor product is isomorphic to the direct sum of the tensor products of the homologies of and is. It turns out that this formula must be extended to the direct sum of the first torsion of the homology groups.

Wording of the sentence

Let and two chain complexes of modules over a principal ideal ring and one of the chain complexes composed exclusively of flat modules. Then for any integer a natural short exact sequence

This sequence is divided, that is isomorphic to a direct sum of the other two components of the sequence, but not in a natural manner.

Importance

The set of Eilenberg - Zilber returns the calculation of the singular homology of a product of topological spaces on the tensor product of the singular homology of the spaces involved. The set of Kiinneth is in this sentence missing algebraic part to perform the calculation of the homology of a product space over.

The universal Koeffiziententheorem

If the chain complex only at the 0 -th digit different from 0 chain complex, so are most summands from the above Künnethformel 0 and we obtain the exact sequence

And that is none other than the universal Koeffiziententheorem.