Recall the tensor product of abelian groups:
Tensor Product of Abelian Groups
Let
and be abelian groups. The tensor product is defined as the abelian group generated by the symbols for , subject to the following relations for all and :
Consequences of the Definition
- Zero Element: The zero element of
is . The relations imply that and for any . - Inverses: The inverse of an element
is , which can be shown to be equal to and . - Integer Multiplication: For any integer
, the relations imply that .
Properties of Tensor Products
Let
, and be abelian groups.
- Commutativity:
. - Distributivity over Direct Sums:
. - Associativity:
. - Identity Element:
. - Tensor Product with Cyclic Groups:
. - Induced Homomorphisms: A pair of homomorphisms
and induces a homomorphism via the rule on generators: - Universal Property: A bilinear map
induces a unique homomorphism from the tensor product into , which sends to .
Cross Product
Cross Product
Let
and be topological spaces, and let be a ring. The cross product is a bilinear map that sends , where and are the projection maps.
Lemma
The cross product is bilinear, so it induces
Künneth Formula for Cohomology
If
and are CW complexes and or is a finitely generated free -module for all , then the cross product is an isomorphism of graded rings.
e.g.
The general Künneth formula relates the (co)homology of a product