Künneth formula

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

(1)

where and are homomorphisms of degree 0 and , respectively (see [2]). 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 [2], [3], 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

where and are homomorphisms of degree 0 and 1, respectively (see [2], [3]).

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

Similar formulas are available for the relative homology and cohomology [3], [4].

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 [3]:

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 [6]. 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 [9]:

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 [7] there exist two spectral sequences and with initial terms

and

having the same limit. The awkward formulation of the Künneth formula assumes a more familiar form in terms of derived functors [11]:

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 [8]), 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 [5]). A particular case of this proposition is the Bott periodicity theorem for complex vector bundles. A Künneth formula is also known in bordism theory [10].

References

[1a]	H. Künneth, "Ueber die Bettische Zahlen einer Produktmannigfaltigkeit" Math. Ann. , 90 (1923) pp. 65–85
[1b]	H. Künneth, "Ueber die Torsionszahlen von Produktmannigfaltigkeiten" Math. Ann. , 91 (1924) pp. 125–134
[2]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) MR0077480 Zbl 0075.24305
[3]	A. Dold, "Lectures on algebraic topology" , Springer (1980) MR0606196 Zbl 0434.55001
[4]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) MR0210112 MR1325242 Zbl 0145.43303
[5]	M.F. Atiyah, "-theory: lectures" , Benjamin (1967) MR224083
[6]	L. Kaup, "Eine Künnethformel für Fréchetgarben" Math. Z. , 97 : 2 (1967) pp. 158–168
[7]	A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" Publ. Math. IHES , 17 (1963) pp. Chapt. 3, Part 2
[8]	M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , Théorie des topos et cohomologie étale des schémas (SGA 4, vol. III) , Lect. notes in math. , 305 , Springer (1973)
[9]	J. Sampson, G. Washnitzer, "A Künneth formula for coherent algebraic sheaves" Illinois J. Math. , 3 : 3 (1959) pp. 389–402
[10]	P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) MR0176478 Zbl 0125.40103
[11]	R. Hartshorne, "Residues and duality" , Springer (1966) MR0222093 Zbl 0212.26101

Comments

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]).

References

[a1]	L. Hodgkin, "The equivariant Künneth theorem in -theory" , Lect. notes in math. , 496 , Springer (1975)