= Künneth theorem
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For chain complexes of free modules over a principal ideal domain $R$, the Künneth theorem gives a natural split short exact sequence
$$
0\to\bigoplus_{i+j=n}H_i(C)\otimes_RH_j(D)\to H_n(C\otimes_RD)\to\bigoplus_{i+j=n-1}\operatorname{Tor}_1^R(H_i(C),H_j(D))\to0,
$$
although the splitting is not natural. Over a field the Tor term vanishes, and the cohomological cross product gives $H^*(X\times Y;k)\cong H^*(X;k)\otimes_kH^*(Y;k)$ as graded rings under standard finiteness hypotheses.
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