Theorem 6.7.8 (Künneth theorem). Let \(R\) be a principal ideal domain and let \(X, Y \in \An \) be animae. For each \(n \in \Z \), there is a natural short exact sequence \[ 0 \to \bigoplus _{i+j=n} H_i(X;R) \otimes _R H_j(Y;R) \to H_n(X \times Y;R) \to \bigoplus _{i+j=n-1} \Tor _1^R(H_i(X;R), H_j(Y;R)) \to 0, \] and this sequence splits, though not naturally.
Proof. By Proposition 6.7.7, we have \(H_n(X \times Y;R) \cong H_n(C_*(X;R) \otimes ^{\bL }_R C_*(Y;R))\). The result now follows immediately from the algebraic Künneth formula (Theorem 6.7.2). □
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