Theorem 6.7.11 (Universal coefficient theorem for homology). Let \(X \in \An \) be an anima and let \(A\) be an abelian group. For each \(n \in \Z \), there is a natural short exact sequence \[ 0 \to H_n(X) \otimes _{\Z } A \to H_n(X;A) \to \Tor _1^{\Z }(H_{n-1}(X), A) \to 0, \] and this sequence splits, though not naturally.
Proof. By Proposition 6.7.10, we have \(H_n(X;A) \cong H_n(C_*(X;\Z ) \otimes ^{\bL } A)\). Viewing \(A\) as a chain complex concentrated in degree \(0\), the statement is thus an instance of the algebraic Künneth formula (Theorem 6.7.2), using that \(H_j(A) = 0\) for \(j \neq 0\) and \(H_0(A) = A\). □
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