Theorem 6.7.13 (Universal coefficient theorem for cohomology). 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 \[ 0 \to \Ext ^1_{\Z }(H_{n-1}(X), A) \to H^n(X;A) \to \Hom _{\Z }(H_n(X), A) \to 0, \] and this sequence splits, though not naturally.
Proof. By Proposition 6.7.12, we have \(H^n(X;A)\cong H_{-n}(\bR \uHom _{\Z }(C_*(X;\Z ),A))\). The result is therefore the case \(R=\Z \), \(M=A\) of Theorem 6.7.3. □
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