Special and general types
A Künneth theorem is a statement relating the homology or cohomology of two objects and with that of their product .
In good situations it identifies the (co)homology of a product with the tesor product of the (co)homologies: . But in general this simple relation receives corrections by Tor-groups.
In ordinary homology
We discuss the Künneth theorem in ordinary homology.
Let be a ring and write Mod for its category of modules. (For instance the integers, in which case Ab is the category of abelian groups. )
For two modules, write for their tensor product of modules. Similarly for two chain complexes of -modules, write for their tensor product of chain complexes. Finally write for the first Tor-module of with .
We discuss the Künneth theorem over in stages, starting with important special cases and then passing to more general statements.
Over a field
Over a principal ideal domain
Over a general ring
All these versions hold for chain homology and tensor products of general chain complexes. But under the Eilenberg-Zilber theorem all these statements apply directly in particular to the singular homology of topological spaces and their products. This is discussed below in
Over a field
Let be a field.
For a field, given two chain complexes of -vector spaces , for each there is an isomorphism
between the chain homology of the tensor product of chain complexes and the tensor product of chain homologies.
For a proof see the proof of theorem 2 below, of which this is a special case.
Over a principal ideal domain
Let now be a ring which is a principal ideal domain. This may be a field as above, or for instance it may be the ring of integers, in which case -modules are equivalently just abelian groups.
For a principal ideal domain, given a chain complex of free modules over and given any other chain complex , then for each there is a short exact sequence of the form
This appears for instance as (Hatcher, theorem 3B.5).
of theorem 2
Notice that since is assumed to be free, hence a direct sum of with itself, since the tensor product of modules distributes over direct sums, and since chain homology respects direct sums, we have
First consider now the special case that all the differentials of are zero, so that . In this case (1) yields and therefore
Since is a free module by assumption, it has no Tor-terms (by the discussion there) and hence this is the statement to be shown.
Now let be a general chain complex of free modules. Notice that for each the cycle-chain-boundary-short exact sequence
splits due to the assumption that is a free module, and hence (as discussed at split exact sequence) that it exhibits a direct sum decomposition . Since the tensor product of modules distributes over direct sum, it follows that tensoring with any yields another short exact sequence
This means that if we regard the graded modules and of chains and of boundaries as chain complexes with zero-differentials, then we have a short exact sequence of chain complexes
This induces its homology long exact sequence of the form
Here the terms involving the complexes and of boundaries and cycles may be evalutated, since these have zero differentials, via the special case discussed at the beginning of this proof to yield the long exact sequence
where is the morphism induced from the inclusion of boundaries into cycles.
This means that by quotienting out an image on the left and a kernel on the right, we obtain a short exact sequence
Since the tensor product of modules is a right exact functor it commutes with the cokernel on the left, as does the formation of direct sums, and so we have
This is the left term in the short exact sequence to be shown. For the right term the analogous argument does not quite go through, because tensoring is not in addition a left exact functor, in general. The failure to be so is precisely measured by the Tor-module:
Notice that by the assumption that is free and using the fact (discussed at principal ideal domain) that over our the submodules are themselves free modules, the defining short exact sequence exhibits a projective resolution of . Therefore by definition of Tor we have
This identifies the term on the right of the exact sequence to be shown.
Over general rings
For any ring, there is a spectral sequence converging to the homology of the tensor product, whose second page involves all the Tor-modules: the Künneth spectral sequence
For instance (Williams, section 4.2).
In Generalised cohomology theories
The Künneth theorem for generalised cohomology theories is a special case of the universal coefficient theorem. Let be a ring spectrum, and two spectra. Then we define a new cohomology theory as the -module spectrum where is the function spectrum (that is, the internal hom in the category of spectra). The (reduced) cohomology of in this cohomology theory is thus given by:
A universal coefficient theorem gives a way of computing from knowledge of and . In this case, so our initial data is and .
For more details, see the page on the universal coefficient theorem.
For singular homology of products of topological spaces
All of these statements have their analogs for singular homology of topological spaces Top: by the Eilenberg-Zilber theorem there is a quasi-isomorphism
between the singular chain complex of the product space and the tensor product of chain complexes of the separate singular chain complexes. Hence in particular there are isomorphisms of singular homology
Using this in the above statements of the Künneth theorem yields directly the Künneth theorem for singular homology of topological spaces.
In ordinary (co)homology
The original articles are
H. Künneth, Über die Bettischen Zahlen einer Produktmannigfaltigkeit Math. Ann. , 90 (1923) pp. 65–85
H. Künneth, Über die Torsionszahlen von Produktmannigfaltigkeiten Math. Ann. , 91 (1924) pp. 125–134
Lecture notes include
- Rob Thompson, Products and the Künneth theorem (pdf)
Section 3.B of
Section 4.2 in
In generalized (co)homology
See the corresponding references at universal coefficient theorem.
In the context of parameterized spectra