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
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then there is an exact sequence of graded modules
![]() | (1) |
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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
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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
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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:
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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:
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where and
are homomorphisms of degree 0 and 1, respectively. For example, if
is a field, then
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and if it is also true that all , or all
, are finite-dimensional, then
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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]:
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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
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where is the symbol for the completed tensor product and
,
are open. If the spaces
and
are separable, one has the Künneth formula
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In particular, coherent analytic sheaves ,
on complex-analytic spaces
,
with countable bases are nuclear and
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where ,
are the analytic inverse images of
and
under the projections
and
. Thus, if
and
are separable, then
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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]:
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Here is the sheaf on
whose modules of sections over
(
is an open affine subset of
,
an open affine subset of
) are
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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
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and
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having the same limit. The awkward formulation of the Künneth formula assumes a more familiar form in terms of derived functors [11]:
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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
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A Künneth formula is also valid for étale sheaves of -modules on schemes
and
, where
is a finite ring. It may be written as
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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
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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
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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) |
[3] | A. Dold, "Lectures on algebraic topology" , Springer (1980) |
[4] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |
[5] | M.F. Atiyah, "![]() |
[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) |
[11] | R. Hartshorne, "Residues and duality" , Springer (1966) |
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 ![]() |
Künneth formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%BCnneth_formula&oldid=23358