A formula expressing the homology (or cohomology) of a tensor product of complexes or a direct product of spaces in terms of the homology (or cohomology) of the factors.
Let be an associative ring with a unit (cf. Associative rings and algebras), and let and be chain complexes of right and left -modules, respectively. Let be the complex associated with the tensor product of and over . If
then there is an exact sequence of graded modules
where and are homomorphisms of degree 0 and , respectively (see ). There is an analogous exact sequence for cochain complexes, with a homomorphism of degree 1. If (e.g. or is a flat -module) and is hereditary, the sequence (1) exists and splits , , so that
This is the Künneth formula; the term Künneth formula (or Künneth relation) is sometimes also applied to the exact sequence (1). There is a generalization of (1) in which the tensor product is replaced by an arbitrary two-place functor , on the category of -modules with values in the same category, that is covariant in and contravariant in . In particular, the functor yields a formula expressing the cohomology , where is a right chain complex and a left cochain complex over , in terms of and . Indeed, if is hereditary and (e.g. is free), one has the split exact sequence
Let , be topological spaces and let , be modules over a principal ideal ring such that . Then the singular homologies of the spaces , , are connected by the following split exact sequence:
where and are homomorphisms of degree 0 and , respectively. If one assumes in addition that either all and , or all and , are finitely generated, an analogous exact sequence is valid for the singular cohomologies:
where and are homomorphisms of degree 0 and 1, respectively. For example, if is a field, then
and if it is also true that all , or all , are finite-dimensional, then
In the case , the module has the structure of a skew tensor product (cf. Skew product) of algebras, with a homomorphism of algebras. Thus, if and all , or all , are finitely generated, one has the following isomorphism of algebras :
If and are finite polyhedra, the Künneth formula enables one to find the Betti numbers and torsion coefficients of the polyhedron in terms of the analogous invariants of and . These are in fact the original results of H. Künneth himself . In particular, if is the -th Betti number of the polyhedron and if
is its Poincaré polynomial, then .
In the theory of cohomology with values in a sheaf there is yet another variant of the Künneth formula . Let and be topological spaces with countable bases, and let and be Fréchet sheaves on and (see Coherent analytic sheaf). Suppose that (or ) is a nuclear sheaf (i.e. is a nuclear space for all open ). Then the Fréchet sheaf is defined on such that
where is the symbol for the completed tensor product and , are open. If the spaces and are separable, one has the Künneth formula
In particular, coherent analytic sheaves , on complex-analytic spaces , with countable bases are nuclear and
where , are the analytic inverse images of and under the projections and . Thus, if and are separable, then
The Künneth formulas also figure in algebraic geometry, usually in the following version. Let and be algebraic varieties over a field , and let and be coherent algebraic sheaves (cf. Coherent algebraic sheaf) on and , respectively. Then :
Here is the sheaf on whose modules of sections over ( is an open affine subset of , an open affine subset of ) are
More generally, let and be morphisms (cf. Morphism) in the category of schemes, let be their fibred product, and let and be quasi-coherent sheaves (cf. Quasi-coherent sheaf) of modules on and . Generalizing the construction of the sheaf , one can introduce sheaves of modules on whose modules of sections for affine , and are isomorphic to , where . Then  there exist two spectral sequences and with initial terms
having the same limit. The awkward formulation of the Künneth formula assumes a more familiar form in terms of derived functors :
If the sheaves and are flat over , then the spectral sequence is degenerate. Similarly, degenerates if all (or all ) are flat over . If both spectral sequences and are degenerate, the Künneth formula becomes
A Künneth formula is also valid for étale sheaves of -modules on schemes and , where is a finite ring. It may be written as
where the means that the cohomology is taken with compact support. In particular (see ), if and are complete algebraic varieties, the Künneth formula for the -adic cohomology is
The formula has been proved for arbitrary varieties only on the assumption that the singularities can be resolved, e.g. for varieties over a field of characteristic zero.
There is also a version of the Künneth formula in -theory. Let be a space such that the group is finitely generated, and let be a cellular space. Then there is an exact sequence of -graded modules
where and are homomorphisms of degree 0 and 1, respectively (see ). A particular case of this proposition is the Bott periodicity theorem for complex vector bundles. A Künneth formula is also known in bordism theory .
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|[1b]||H. Künneth, "Ueber die Torsionszahlen von Produktmannigfaltigkeiten" Math. Ann. , 91 (1924) pp. 125–134|
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More generally, cohomology theories have a Künneth formula spectral sequence for , where and are as in the last section of the main article above (e.g., for equivariant -theory see [a1]).
|[a1]||L. Hodgkin, "The equivariant Künneth theorem in -theory" , Lect. notes in math. , 496 , Springer (1975)|
Künneth formula. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=K%C3%BCnneth_formula&oldid=24488