 Künneth theorem

For chain complexes of free modules over a principal ideal domain $R$, the Künneth theorem gives a natural split short exact sequencealthough the splitting is not natural. Over a field the Tor term vanishes, and the cohomological cross product gives $H^{\ast }(X\times Y;k)\cong H^{\ast }(X;k){\otimes }_k{H}^{\ast }(Y;k)$ as graded rings under standard finiteness hypotheses.

The Künneth theorem is an important result in algebraic topology that relates the homology groups of a product of two topological spaces to the homology groups of the individual spaces. It is particularly useful in the computation of homology groups for spaces that can be expressed as products of simpler spaces.