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====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> H. Künneth,   "Ueber die Bettische Zahlen einer Produktmannigfaltigkeit" ''Math. Ann.'' , '''90''' (1923) pp. 65–85</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> H. Künneth,   "Ueber die Torsionszahlen von Produktmannigfaltigkeiten" ''Math. Ann.'' , '''91''' (1924) pp. 125–134</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Cartan,   S. Eilenberg,   "Homological algebra" , Princeton Univ. Press (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Dold,   "Lectures on algebraic topology" , Springer (1980)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.H. Spanier,   "Algebraic topology" , McGraw-Hill (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.F. Atiyah,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010185.png" />-theory: lectures" , Benjamin (1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> L. Kaup,   "Eine Künnethformel für Fréchetgarben" ''Math. Z.'' , '''97''' : 2 (1967) pp. 158–168</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A. Grothendieck,   J. Dieudonné,   "Eléments de géometrie algébrique" ''Publ. Math. IHES'' , '''17''' (1963) pp. Chapt. 3, Part 2</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> J. Sampson,   G. Washnitzer,   "A Künneth formula for coherent algebraic sheaves" ''Illinois J. Math.'' , '''3''' : 3 (1959) pp. 389–402</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> P.E. Conner,   E.E. Floyd,   "Differentiable periodic maps" , Springer (1964)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> R. Hartshorne,   "Residues and duality" , Springer (1966)</TD></TR></table>
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<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> H. Künneth, "Ueber die Bettische Zahlen einer Produktmannigfaltigkeit" ''Math. Ann.'' , '''90''' (1923) pp. 65–85</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> H. Künneth, "Ueber die Torsionszahlen von Produktmannigfaltigkeiten" ''Math. Ann.'' , '''91''' (1924) pp. 125–134</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) {{MR|0077480}} {{ZBL|0075.24305}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1980) {{MR|0606196}} {{ZBL|0434.55001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.F. Atiyah, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010185.png" />-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> L. Kaup, "Eine Künnethformel für Fréchetgarben" ''Math. Z.'' , '''97''' : 2 (1967) pp. 158–168</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" ''Publ. Math. IHES'' , '''17''' (1963) pp. Chapt. 3, Part 2</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> J. Sampson, G. Washnitzer, "A Künneth formula for coherent algebraic sheaves" ''Illinois J. Math.'' , '''3''' : 3 (1959) pp. 389–402</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) {{MR|0176478}} {{ZBL|0125.40103}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> R. Hartshorne, "Residues and duality" , Springer (1966) {{MR|0222093}} {{ZBL|0212.26101}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hodgkin,   "The equivariant Künneth theorem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010190.png" />-theory" , ''Lect. notes in math.'' , '''496''' , Springer (1975)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hodgkin, "The equivariant Künneth theorem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010190.png" />-theory" , ''Lect. notes in math.'' , '''496''' , Springer (1975)</TD></TR></table>

Revision as of 16:59, 15 April 2012

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)
How to Cite This Entry:
Künneth formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%BCnneth_formula&oldid=23358
This article was adapted from an original article by V.I. DanilovA.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article