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 $\Lambda$ 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)