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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  $  \Lambda $
be an associative ring with a unit (cf. [[Associative rings and algebras|Associative rings and algebras]]), and let  $  A $
and  $  C $
be chain complexes of right and left  $  \Lambda $-
modules, respectively. Let  $  A \otimes C $
be the complex associated with the [[Tensor product|tensor product]] of  $  A $
and  $  C $
over  $  \Lambda $. 
If

$$ 
 \mathop{\rm Tor} _ {1} ( B ( A), B ( C))  = \ 
 \mathop{\rm Tor} _ {1} ( H _ {*} ( A), B ( C)) =
$$

$$ 
= \ 
 \mathop{\rm Tor} _ {1} ( B ( A), Z ( C))  =   \mathop{\rm Tor} _ {1} ( H _ {*} ( A), Z ( C))  =  0,
$$

then there is an exact sequence of graded modules

$$ \tag{1 }
0  \rightarrow  H _ {*} ( A) \otimes H _ {*} ( C)   \mathop \rightarrow \limits ^  \alpha   \ 
H _ {*} ( A \otimes C)   \mathop \rightarrow \limits ^  \beta   \ 
$$

$$ 
 \mathop \rightarrow \limits ^  \beta     \mathop{\rm Tor} _ {1} ( H _ {*} ( A), H _ {*} ( C))  \rightarrow  0,
$$

where  $  \alpha $
and  $  \beta $
are homomorphisms of degree 0 and  $  - 1 $, 
respectively (see [[#References|[2]]]). There is an analogous exact sequence for cochain complexes, with a homomorphism  $  \beta $
of degree 1. If  $  H _ {*} (  \mathop{\rm Tor} _ {1} ( A, C)) = 0 $(
e.g.  $  A $
or  $  C $
is a flat  $  \Lambda $-
module) and  $  \Lambda $
is hereditary, the sequence (1) exists and splits [[#References|[2]]], [[#References|[3]]], so that

$$ 
H _ {n} ( A \otimes C)  = \ 
\sum _ {p + q = n } H _ {p} ( A) \otimes H _ {q} ( C) +
$$

$$ 
+ 
\sum _ {p + q = n - 1 }  \mathop{\rm Tor} _ {1} ( H _ {p} ( A), H _ {q} ( C)).
$$

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  $  T( A, C) $, 
on the category of  $  \Lambda $-
modules with values in the same category, that is covariant in  $  A $
and contravariant in  $  C $. 
In particular, the functor  $  T ( A, C) =  \mathop{\rm Hom} ( A, C) $
yields a formula expressing the cohomology  $  H  ^ {*} (  \mathop{\rm Hom} ( A, C)) $, 
where  $  A $
is a right chain complex and  $  C $
a left cochain complex over  $  \Lambda $, 
in terms of  $  H _ {*} ( A) $
and  $  H  ^ {*} ( C) $. 
Indeed, if  $  \Lambda $
is hereditary and  $  H  ^ {*} (  \mathop{\rm Ext}  ^ {1} ( A, C) ) = 0 $(
e.g.  $  A $
is free), one has the split exact sequence

$$ 
0  \rightarrow   \mathop{\rm Ext}  ^ {1} ( H _ {*} ( A),\ 
H  ^ {*} ( C))  \rightarrow ^ { {\beta  ^  \prime } } \ 
H  ^ {*} (  \mathop{\rm Hom} ( A, C))  \rightarrow ^ { {\alpha  ^  \prime } } \ 
$$

$$ 
\rightarrow ^ { {\alpha  ^  \prime } }   \mathop{\rm Hom} ( H _ {*} ( A), H  ^ {*} ( C))  \rightarrow  0,
$$

where  $  \alpha  ^  \prime  $
and  $  \beta  ^  \prime  $
are homomorphisms of degree 0 and 1, respectively (see [[#References|[2]]], [[#References|[3]]]).

Let  $  X $, 
$  Y $
be topological spaces and let  $  L $, 
$  M $
be modules over a principal ideal ring  $  R $
such that  $   \mathop{\rm Tor} _ {1} ( L, M) = 0 $. 
Then the singular homologies of the spaces  $  X $, 
$  Y $, 
$  X \times Y $
are connected by the following split exact sequence:

$$ 
0  \rightarrow  H _ {*} ( X, L) \otimes H _ {*} ( Y, M)   \mathop \rightarrow \limits ^  \alpha  
 H _ {*} ( X \times Y, L \otimes M)   \mathop \rightarrow \limits ^  \beta  
$$

$$ 
 \mathop \rightarrow \limits ^  \beta     \mathop{\rm Tor} _ {1} ( H _ {*} ( X, L), H _ {*} ( Y, M))  \rightarrow  0,
$$

where  $  \alpha $
and  $  \beta $
are homomorphisms of degree 0 and  $  - 1 $, 
respectively. If one assumes in addition that either all  $  H _ {k} ( X, R) $
and  $  H _ {k} ( Y, R) $, 
or all  $  H _ {k} ( Y, R) $
and  $  M $, 
are finitely generated, an analogous exact sequence is valid for the singular cohomologies:

$$ 
0  \rightarrow  H  ^ {*} ( X, L) \otimes H  ^ {*} ( Y, M)   \mathop \rightarrow \limits ^  \alpha   \ 
H  ^ {*} ( X \times Y, L \otimes M)   \mathop \rightarrow \limits ^  \beta  
$$

$$ 
 \mathop \rightarrow \limits ^  \beta     \mathop{\rm Tor} _ {1} ( H  ^ {*} ( X, L), H  ^ {*} ( Y, M))  \rightarrow  0,
$$

where  $  \alpha $
and  $  \beta $
are homomorphisms of degree 0 and 1, respectively. For example, if  $  R $
is a field, then

$$ 
H _ {*} ( X \times Y, R)  \cong \ 
H _ {*} ( X, R) \otimes H _ {*} ( Y, R),
$$

and if it is also true that all  $  H _ {k} ( X, R) $, 
or all  $  H _ {k} ( Y, R) $, 
are finite-dimensional, then

$$ 
H  ^ {*} ( X \times Y, R)  \cong \ 
H  ^ {*} ( X, R) \otimes H  ^ {*} ( Y, R).
$$

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

In the case  $  L = M = R $, 
the module  $  H  ^ {*} ( X, R) \otimes H  ^ {*} ( Y, R) $
has the structure of a skew tensor product (cf. [[Skew product|Skew product]]) of algebras, with  $  \alpha $
a homomorphism of algebras. Thus, if  $   \mathop{\rm Tor} _ {1} ( H  ^ {*} ( X, R), H  ^ {*} ( Y, R)) = 0 $
and all  $  H _ {k} ( X, R) $, 
or all  $  H _ {k} ( Y, R) $, 
are finitely generated, one has the following isomorphism of algebras [[#References|[3]]]:

$$ 
H  ^ {*} ( X \times Y, R)  \cong \ 
H  ^ {*} ( X, R) \otimes H  ^ {*} ( Y, R).
$$

If  $  X $
and  $  Y $
are finite polyhedra, the Künneth formula enables one to find the Betti numbers and torsion coefficients of the polyhedron  $  X \times Y $
in terms of the analogous invariants of  $  X $
and  $  Y $. 
These are in fact the original results of H. Künneth himself . In particular, if  $  b _ {k} ( X) $
is the  $  k $-
th Betti number of the polyhedron  $  X $
and if

$$ 
p ( X)  =  \sum _ {k \geq  0 } b _ {k} ( X) t  ^ {k}
$$

is its Poincaré polynomial, then  $  p ( X \times Y) = p ( X) p ( Y) $.

In the theory of cohomology with values in a sheaf there is yet another variant of the Künneth formula [[#References|[6]]]. Let  $  X $
and  $  Y $
be topological spaces with countable bases, and let  $  {\mathcal F} $
and  $  {\mathcal G} $
be Fréchet sheaves on  $  X $
and  $  Y $(
see [[Coherent analytic sheaf|Coherent analytic sheaf]]). Suppose that  $  {\mathcal F} $(
or  $  {\mathcal G} $) 
is a nuclear sheaf (i.e.  $  {\mathcal F} ( U) $
is a nuclear space for all open  $  U \subset  X $). 
Then the Fréchet sheaf  $  {\mathcal F} \widetilde \otimes   {\mathcal G} $
is defined on  $  X \times Y $
such that

$$ 
( {\mathcal F} \widetilde \otimes   {\mathcal G} ) ( U \times V)  = \ 
{\mathcal F} ( U) \widetilde \otimes   {\mathcal G} ( V),
$$

where  $  \widetilde \otimes   $
is the symbol for the completed tensor product and  $  U \subset  X $, 
$  V \subset  Y $
are open. If the spaces  $  H  ^ {*} ( X, {\mathcal F} ) $
and  $  H  ^ {*} ( Y, {\mathcal G} ) $
are separable, one has the Künneth formula

$$ 
H  ^ {*} ( X \times Y, {\mathcal F} \widetilde \otimes   {\mathcal G} )  \cong \ 
H  ^ {*} ( X, {\mathcal F} ) \widetilde \otimes   H  ^ {*} ( Y, {\mathcal G} ).
$$

In particular, coherent analytic sheaves  $  {\mathcal F} $, 
$  {\mathcal G} $
on complex-analytic spaces  $  X $, 
$  Y $
with countable bases are nuclear and

$$ 
{\mathcal F} \widetilde \otimes   {\mathcal G}  \cong \ 
{\mathcal F}  ^ {*} \otimes _ { {\mathcal O} _ {X \times Y }  } {\mathcal G}  ^ {*} ,
$$

where  $  {\mathcal F}  ^ {*} $, 
$  {\mathcal G}  ^ {*} $
are the analytic inverse images of  $  {\mathcal F} $
and  $  {\mathcal G} $
under the projections  $  X \times Y \rightarrow X $
and  $  X \times Y \rightarrow Y $. 
Thus, if  $  H  ^ {*} ( X, {\mathcal F} ) $
and  $  H  ^ {*} ( Y, {\mathcal G} ) $
are separable, then

$$ 
H  ^ {*} \left ( X \times Y, {\mathcal F}  ^ {*} \otimes
_ { {\mathcal O} _ {X \times Y }  }
{\mathcal G}  ^ {*} \right )  \cong \ 
H  ^ {*} ( X, {\mathcal F} ) \widetilde \otimes   H  ^ {*} ( Y, {\mathcal G} ).
$$

The Künneth formulas also figure in algebraic geometry, usually in the following version. Let  $  X $
and  $  Y $
be algebraic varieties over a field  $  k $, 
and let  $  {\mathcal F} $
and  $  {\mathcal G} $
be coherent algebraic sheaves (cf. [[Coherent algebraic sheaf|Coherent algebraic sheaf]]) on  $  X $
and  $  Y $, 
respectively. Then [[#References|[9]]]:

$$ 
H  ^ {*} \left ( X \times Y, {\mathcal F} \otimes _ { k } {\mathcal G} \right )  \cong \ 
H  ^ {*} ( X, {\mathcal F} ) \otimes _ { k } H  ^ {*} ( Y, {\mathcal G} ).
$$

Here  $  {\mathcal F} \otimes _ {k} {\mathcal G} $
is the sheaf on  $  X \times Y $
whose modules of sections over  $  U \times V $(
$  U $
is an open affine subset of  $  X $, 
$  V $
an open affine subset of  $  Y $) 
are

$$ 
\Gamma ( U, {\mathcal F} ) \otimes _ { k } \Gamma ( V, {\mathcal G} ).
$$

More generally, let  $  p:  X \rightarrow S $
and  $  q:  Y \rightarrow S $
be morphisms (cf. [[Morphism|Morphism]]) in the category of schemes, let  $  h:  X \times _ {S} Y \rightarrow S $
be their fibred product, and let  $  {\mathcal F} $
and  $  {\mathcal G} $
be quasi-coherent sheaves (cf. [[Quasi-coherent sheaf|Quasi-coherent sheaf]]) of modules on  $  X $
and  $  Y $. 
Generalizing the construction of the sheaf  $  {\mathcal F} \otimes _ {k} {\mathcal G} $, 
one can introduce sheaves of modules  $   \mathop{\rm Tor} _ {m}  ^ {S} ( {\mathcal F} , {\mathcal G} ) $
on  $  X \times Y $
whose modules of sections for affine  $  S $, 
$  X $
and  $  Y $
are isomorphic to  $   \mathop{\rm Tor} _ {m}  ^ {A} ( \Gamma ( X, {\mathcal F} ), \Gamma ( Y, {\mathcal G} )) $, 
where  $  A = \Gamma ( S, {\mathcal O} _ {S} ) $. 
Then [[#References|[7]]] there exist two spectral sequences  $  ( E  ^ {r} ) $
and  $  ( {}  ^  \prime  E  ^ {r} ) $
with initial terms

$$ 
E _ {n, m }   ^ {2}  = \ 
R  ^ {-} n h _ {*} (  \mathop{\rm Tor} _ {m}  ^ {S} ( {\mathcal F} , {\mathcal G} ))
$$

and

$$ 
E _ {n, m }   ^  \prime  2  = \ 
\oplus _ {m _ {1} + m _ {2} = m }
 \mathop{\rm Tor} _ {n}  ^ {S}
( R ^ {- m _ {1} } p _ {*} {\mathcal F} , R ^ {- m _ {2} } q _ {*} {\mathcal G} ),
$$

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

$$ 
Rp _ {*} ( {\mathcal F} )
\otimes _ { {\mathcal O} _ {S} } ^ { L } 
Rq _ {*} ( {\mathcal G} )  = \ 
Rh _ {*} \left (
{\mathcal F} \otimes _ { {\mathcal O} _ {S} } ^ { L } 
{\mathcal G} \right ) .
$$

If the sheaves  $  {\mathcal F} $
and  $  {\mathcal G} $
are flat over  $  S $, 
then the spectral sequence  $  ( E  ^ {r} ) $
is degenerate. Similarly,  $  ( {}  ^  \prime  E  ^ {r} ) $
degenerates if all  $  R  ^ {k} p _ {*} ( {\mathcal F} ) $(
or all  $  R  ^ {k} q _ {*} ( {\mathcal G} ) $) 
are flat over  $  S $. 
If both spectral sequences  $  ( E  ^ {r} ) $
and  $  ( {}  ^  \prime  E  ^ {r} ) $
are degenerate, the Künneth formula becomes

$$ 
R  ^ {*} h _ {*} \left ( {\mathcal F} \otimes _ { {\mathcal O} _ {S} } {\mathcal G} \right )  \cong \ 
R  ^ {*} p _ {*} ( {\mathcal F} ) \otimes _ { {\mathcal O} _ {S} }
R  ^ {*} q _ {k} ( {\mathcal G} ).
$$

A Künneth formula is also valid for étale sheaves of  $  A $-
modules on schemes  $  X $
and  $  Y $, 
where  $  A $
is a finite ring. It may be written as

$$ 
Rp _ {!} ( {\mathcal F} ) \otimes _ { A } ^ { L }  Rq _ {!} ( {\mathcal G} )  = \ 
Rh _ {!} \left ( {\mathcal F} \otimes _ { A } ^ { L }  {\mathcal G} \right ) ,
$$

where the  $  ! $
means that the cohomology is taken with compact support. In particular (see [[#References|[8]]]), if  $  X $
and  $  Y $
are complete algebraic varieties, the Künneth formula for the  $  l $-
adic cohomology is

$$ 
H  ^ {*} ( X \times Y, \mathbf Q _ {l} )  = \ 
H  ^ {*} ( X, \mathbf Q _ {l} )
\otimes _ {\mathbf Q _ {l} }
H  ^ {*} ( Y, \mathbf Q _ {l} ).
$$

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 [[K-theory| $  K $-
theory]]. Let  $  X $
be a space such that the group  $  K  ^ {*} ( X) $
is finitely generated, and let  $  Y $
be a cellular space. Then there is an exact sequence of  $  \mathbf Z _ {2} $-
graded modules

$$ 
0  \rightarrow  K  ^ {*} ( X) \otimes K  ^ {*} ( Y)   \mathop \rightarrow \limits ^  \alpha  
 K  ^ {*} ( X \times Y)   \mathop \rightarrow \limits ^  \beta  
$$

$$ 
 \mathop \rightarrow \limits ^  \beta     \mathop{\rm Tor} _ {1} ( K  ^ {*} ( X), K  ^ {*} ( Y))  \rightarrow  0,
$$

where  $  \alpha $
and  $  \beta $
are homomorphisms of degree 0 and 1, respectively (see [[#References|[5]]]). A particular case of this proposition is the [[Bott periodicity theorem|Bott periodicity theorem]] for complex vector bundles. A Künneth formula is also known in bordism theory [[#References|[10]]].

====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) {{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 géométrie 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>

====Comments====
More generally, cohomology theories have a Künneth formula spectral sequence for  $  h  ^ {*} ( X \times Y) $, 
where  $  X $
and  $  Y $
are as in the last section of the main article above (e.g., for equivariant  $  K $-
theory see [[#References|[a1]]]).

====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>