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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k0560101.png" /> be an associative ring with a unit (cf. [[Associative rings and algebras|Associative rings and algebras]]), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k0560102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k0560103.png" /> be chain complexes of right and left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k0560104.png" />-modules, respectively. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k0560105.png" /> be the complex associated with the [[Tensor product|tensor product]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k0560106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k0560107.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k0560108.png" />. If
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k0560109.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Tor} _ {1} ( B ( A), B ( C))  = \
 +
\mathop{\rm Tor} _ {1} ( H _ {*} ( A), B ( C)) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601010.png" /></td> </tr></table>
+
$$
 +
= \
 +
\mathop{\rm Tor} _ {1} ( B ( A), Z ( C))  =   \mathop{\rm Tor} _ {1} ( H _ {*} ( A), Z ( C))  = 0,
 +
$$
  
 
then there is an exact sequence of graded modules
 
then there is an exact sequence of graded modules
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
0 \rightarrow  H _ {*} ( A) \otimes H _ {*} ( C)  \mathop \rightarrow \limits ^  \alpha  \
 +
H _ {*} ( A \otimes C)   \mathop \rightarrow \limits ^  \beta  \
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601012.png" /></td> </tr></table>
+
$$
 +
\mathop \rightarrow \limits ^  \beta    \mathop{\rm Tor} _ {1} ( H _ {*} ( A), H _ {*} ( C))  \rightarrow  0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601014.png" /> are homomorphisms of degree 0 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601015.png" />, respectively (see [[#References|[2]]]). There is an analogous exact sequence for cochain complexes, with a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601016.png" /> of degree 1. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601017.png" /> (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601018.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601019.png" /> is a flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601020.png" />-module) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601021.png" /> is hereditary, the sequence (1) exists and splits [[#References|[2]]], [[#References|[3]]], so that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601022.png" /></td> </tr></table>
+
$$
 +
H _ {n} ( A \otimes C)  = \
 +
\sum _ {p + q = n } H _ {p} ( A) \otimes H _ {q} ( C) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601023.png" /></td> </tr></table>
+
$$
 +
+
 +
\sum _ {p + q = n - 1 }  \mathop{\rm Tor} _ {1} ( H _ {p} ( A), H _ {q} ( C)).
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601024.png" />, on the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601025.png" />-modules with values in the same category, that is covariant in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601026.png" /> and contravariant in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601027.png" />. In particular, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601028.png" /> yields a formula expressing the cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601030.png" /> is a right chain complex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601031.png" /> a left cochain complex over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601032.png" />, in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601034.png" />. Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601035.png" /> is hereditary and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601036.png" /> (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601037.png" /> is free), one has the split exact sequence
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601038.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  \mathop{\rm Ext}  ^ {1} ( H _ {*} ( A),\
 +
H  ^ {*} ( C))  \rightarrow ^ { {\beta  ^  \prime } } \
 +
H  ^ {*} (  \mathop{\rm Hom} ( A, C))  \rightarrow ^ { {\alpha  ^  \prime } } \
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601039.png" /></td> </tr></table>
+
$$
 +
\rightarrow ^ { {\alpha  ^  \prime } }  \mathop{\rm Hom} ( H _ {*} ( A), H  ^ {*} ( C))  \rightarrow  0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601041.png" /> are homomorphisms of degree 0 and 1, respectively (see [[#References|[2]]], [[#References|[3]]]).
+
where $  \alpha  ^  \prime  $
 +
and $  \beta  ^  \prime  $
 +
are homomorphisms of degree 0 and 1, respectively (see [[#References|[2]]], [[#References|[3]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601043.png" /> be topological spaces and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601045.png" /> be modules over a principal ideal ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601046.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601047.png" />. Then the singular homologies of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601050.png" /> are connected by the following split exact sequence:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601051.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  H _ {*} ( X, L) \otimes H _ {*} ( Y, M)  \mathop \rightarrow \limits ^  \alpha 
 +
H _ {*} ( X \times Y, L \otimes M)  \mathop \rightarrow \limits ^  \beta 
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601052.png" /></td> </tr></table>
+
$$
 +
\mathop \rightarrow \limits ^  \beta    \mathop{\rm Tor} _ {1} ( H _ {*} ( X, L), H _ {*} ( Y, M))  \rightarrow  0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601054.png" /> are homomorphisms of degree 0 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601055.png" />, respectively. If one assumes in addition that either all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601057.png" />, or all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601059.png" />, are finitely generated, an analogous exact sequence is valid for the singular cohomologies:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601060.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  H  ^ {*} ( X, L) \otimes H  ^ {*} ( Y, M)  \mathop \rightarrow \limits ^  \alpha  \
 +
H  ^ {*} ( X \times Y, L \otimes M)  \mathop \rightarrow \limits ^  \beta 
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601061.png" /></td> </tr></table>
+
$$
 +
\mathop \rightarrow \limits ^  \beta    \mathop{\rm Tor} _ {1} ( H  ^ {*} ( X, L), H  ^ {*} ( Y, M))  \rightarrow  0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601063.png" /> are homomorphisms of degree 0 and 1, respectively. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601064.png" /> is a field, then
+
where $  \alpha $
 +
and $  \beta $
 +
are homomorphisms of degree 0 and 1, respectively. For example, if $  R $
 +
is a field, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601065.png" /></td> </tr></table>
+
$$
 +
H _ {*} ( X \times Y, R)  \cong \
 +
H _ {*} ( X, R) \otimes H _ {*} ( Y, R),
 +
$$
  
and if it is also true that all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601066.png" />, or all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601067.png" />, are finite-dimensional, then
+
and if it is also true that all $  H _ {k} ( X, R) $,
 +
or all $  H _ {k} ( Y, R) $,  
 +
are finite-dimensional, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601068.png" /></td> </tr></table>
+
$$
 +
H  ^ {*} ( X \times Y, R)  \cong \
 +
H  ^ {*} ( X, R) \otimes H  ^ {*} ( Y, R).
 +
$$
  
 
Similar formulas are available for the relative homology and cohomology [[#References|[3]]], [[#References|[4]]].
 
Similar formulas are available for the relative homology and cohomology [[#References|[3]]], [[#References|[4]]].
  
In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601069.png" />, the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601070.png" /> has the structure of a skew tensor product (cf. [[Skew product|Skew product]]) of algebras, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601071.png" /> a homomorphism of algebras. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601072.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601073.png" />, or all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601074.png" />, are finitely generated, one has the following isomorphism of algebras [[#References|[3]]]:
+
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]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601075.png" /></td> </tr></table>
+
$$
 +
H  ^ {*} ( X \times Y, R)  \cong \
 +
H  ^ {*} ( X, R) \otimes H  ^ {*} ( Y, R).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601077.png" /> are finite polyhedra, the Künneth formula enables one to find the Betti numbers and torsion coefficients of the polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601078.png" /> in terms of the analogous invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601080.png" />. These are in fact the original results of H. Künneth himself . In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601081.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601082.png" />-th Betti number of the polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601083.png" /> and if
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601084.png" /></td> </tr></table>
+
$$
 +
p ( X)  = \sum _ {k \geq  0 } b _ {k} ( X) t  ^ {k}
 +
$$
  
is its Poincaré polynomial, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601085.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601087.png" /> be topological spaces with countable bases, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601089.png" /> be Fréchet sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601091.png" /> (see [[Coherent analytic sheaf|Coherent analytic sheaf]]). Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601092.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601093.png" />) is a nuclear sheaf (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601094.png" /> is a nuclear space for all open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601095.png" />). Then the Fréchet sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601096.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601097.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601098.png" /></td> </tr></table>
+
$$
 +
( {\mathcal F} \widetilde \otimes  {\mathcal G} ) ( U \times V)  = \
 +
{\mathcal F} ( U) \widetilde \otimes  {\mathcal G} ( V),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k05601099.png" /> is the symbol for the completed tensor product and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010101.png" /> are open. If the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010103.png" /> are separable, one has the Künneth formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010104.png" /></td> </tr></table>
+
$$
 +
H  ^ {*} ( X \times Y, {\mathcal F} \widetilde \otimes  {\mathcal G} )  \cong \
 +
H  ^ {*} ( X, {\mathcal F} ) \widetilde \otimes  H  ^ {*} ( Y, {\mathcal G} ).
 +
$$
  
In particular, coherent analytic sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010106.png" /> on complex-analytic spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010108.png" /> with countable bases are nuclear and
+
In particular, coherent analytic sheaves $  {\mathcal F} $,  
 +
$  {\mathcal G} $
 +
on complex-analytic spaces $  X $,  
 +
$  Y $
 +
with countable bases are nuclear and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010109.png" /></td> </tr></table>
+
$$
 +
{\mathcal F} \widetilde \otimes  {\mathcal G}  \cong \
 +
{\mathcal F}  ^ {*} \otimes _ { {\mathcal O} _ {X \times Y }  } {\mathcal G}  ^ {*} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010111.png" /> are the analytic inverse images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010113.png" /> under the projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010115.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010117.png" /> are separable, then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010118.png" /></td> </tr></table>
+
$$
 +
H  ^ {*} \left ( X \times Y, {\mathcal F}  ^ {*} \otimes
 +
_ { {\mathcal O} _ {X \times Y }  }
 +
{\mathcal G}  ^ {*} \right )  \cong \
 +
H  ^ {*} ( X, {\mathcal F} ) \widetilde \otimes  H  ^ {*} ( Y, {\mathcal G} ).
 +
$$
  
The Künneth formulas also figure in algebraic geometry, usually in the following version. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010119.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010120.png" /> be algebraic varieties over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010121.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010123.png" /> be coherent algebraic sheaves (cf. [[Coherent algebraic sheaf|Coherent algebraic sheaf]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010125.png" />, respectively. Then [[#References|[9]]]:
+
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]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010126.png" /></td> </tr></table>
+
$$
 +
H  ^ {*} \left ( X \times Y, {\mathcal F} \otimes _ { k } {\mathcal G} \right )  \cong \
 +
H  ^ {*} ( X, {\mathcal F} ) \otimes _ { k } H  ^ {*} ( Y, {\mathcal G} ).
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010127.png" /> is the sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010128.png" /> whose modules of sections over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010129.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010130.png" /> is an open affine subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010132.png" /> an open affine subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010133.png" />) are
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010134.png" /></td> </tr></table>
+
$$
 +
\Gamma ( U, {\mathcal F} ) \otimes _ { k } \Gamma ( V, {\mathcal G} ).
 +
$$
  
More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010135.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010136.png" /> be morphisms (cf. [[Morphism|Morphism]]) in the category of schemes, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010137.png" /> be their fibred product, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010138.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010139.png" /> be quasi-coherent sheaves (cf. [[Quasi-coherent sheaf|Quasi-coherent sheaf]]) of modules on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010140.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010141.png" />. Generalizing the construction of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010142.png" />, one can introduce sheaves of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010143.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010144.png" /> whose modules of sections for affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010145.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010146.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010147.png" /> are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010148.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010149.png" />. Then [[#References|[7]]] there exist two spectral sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010150.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010151.png" /> with initial terms
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010152.png" /></td> </tr></table>
+
$$
 +
E _ {n, m }  ^ {2}  = \
 +
R  ^ {-} n h _ {*} (  \mathop{\rm Tor} _ {m}  ^ {S} ( {\mathcal F} , {\mathcal G} ))
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010153.png" /></td> </tr></table>
+
$$
 +
E _ {n, m }  ^  \prime  2  = \
 +
\oplus _ {m _ {1} + m _ {2} = m }
 +
\mathop{\rm Tor} _ {n}  ^ {S}
 +
( R ^ {- m _ {1} } p _ {*} {\mathcal F} , R ^ {- m _ {2} } q _ {*} {\mathcal G} ),
 +
$$
  
 
having the same limit. The awkward formulation of the Künneth formula assumes a more familiar form in terms of derived functors [[#References|[11]]]:
 
having the same limit. The awkward formulation of the Künneth formula assumes a more familiar form in terms of derived functors [[#References|[11]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010154.png" /></td> </tr></table>
+
$$
 +
Rp _ {*} ( {\mathcal F} )
 +
\otimes _ { {\mathcal O} _ {S} } ^ { L }
 +
Rq _ {*} ( {\mathcal G} )  = \
 +
Rh _ {*} \left (
 +
{\mathcal F} \otimes _ { {\mathcal O} _ {S} } ^ { L }
 +
{\mathcal G} \right ) .
 +
$$
  
If the sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010156.png" /> are flat over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010157.png" />, then the spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010158.png" /> is degenerate. Similarly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010159.png" /> degenerates if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010160.png" /> (or all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010161.png" />) are flat over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010162.png" />. If both spectral sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010163.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010164.png" /> are degenerate, the Künneth formula becomes
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010165.png" /></td> </tr></table>
+
$$
 +
R  ^ {*} h _ {*} \left ( {\mathcal F} \otimes _ { {\mathcal O} _ {S} } {\mathcal G} \right )  \cong \
 +
R  ^ {*} p _ {*} ( {\mathcal F} ) \otimes _ { {\mathcal O} _ {S} }
 +
R  ^ {*} q _ {k} ( {\mathcal G} ).
 +
$$
  
A Künneth formula is also valid for étale sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010166.png" />-modules on schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010167.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010168.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010169.png" /> is a finite ring. It may be written as
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010170.png" /></td> </tr></table>
+
$$
 +
Rp _ {!} ( {\mathcal F} ) \otimes _ { A } ^ { L }  Rq _ {!} ( {\mathcal G} )  = \
 +
Rh _ {!} \left ( {\mathcal F} \otimes _ { A } ^ { L }  {\mathcal G} \right ) ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010171.png" /> means that the cohomology is taken with compact support. In particular (see [[#References|[8]]]), if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010172.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010173.png" /> are complete algebraic varieties, the Künneth formula for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010174.png" />-adic cohomology is
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010175.png" /></td> </tr></table>
+
$$
 +
H  ^ {*} ( X \times Y, \mathbf Q _ {l} )  = \
 +
H  ^ {*} ( X, \mathbf Q _ {l} )
 +
\otimes _ {\mathbf Q _ {l} }
 +
H  ^ {*} ( Y, \mathbf Q _ {l} ).
 +
$$
  
 
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.
 
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|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010176.png" />-theory]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010177.png" /> be a space such that the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010178.png" /> is finitely generated, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010179.png" /> be a cellular space. Then there is an exact sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010180.png" />-graded modules
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010181.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  K  ^ {*} ( X) \otimes K  ^ {*} ( Y)  \mathop \rightarrow \limits ^  \alpha 
 +
K  ^ {*} ( X \times Y)  \mathop \rightarrow \limits ^  \beta 
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010182.png" /></td> </tr></table>
+
$$
 +
\mathop \rightarrow \limits ^  \beta    \mathop{\rm Tor} _ {1} ( K  ^ {*} ( X), K  ^ {*} ( Y))  \rightarrow  0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010183.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010184.png" /> 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]]].
+
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====
 
====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 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>
+
<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, "$K$-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====
 
====Comments====
More generally, cohomology theories have a Künneth formula spectral sequence for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010186.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010187.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010188.png" /> are as in the last section of the main article above (e.g., for equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k056/k056010/k056010189.png" />-theory see [[#References|[a1]]]).
+
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====
 
====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>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hodgkin, "The equivariant Künneth theorem in $K$-theory" , ''Lect. notes in math.'' , '''496''' , Springer (1975)</TD></TR>
 +
</table>

Latest revision as of 05:56, 19 July 2024


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), 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 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 [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 [2], [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 [2], [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 [3], [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) 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 [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 [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). 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) on $ X $ and $ Y $, respectively. Then [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) 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) 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 [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 [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 [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. 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 [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, "$K$-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 géométrie 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 $ 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 [a1]).

References

[a1] L. Hodgkin, "The equivariant Künneth theorem in $K$-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=24488
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