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