Universal coefficient theorem for cohomology

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For coefficients in an abelian group

Suppose M is an abelian group and X is a space with homology of finite type. The universal coefficients theorem relates the cohomology groups for X with integral coefficients (i.e., with coefficients in \mathbb{Z}) to the cohomology groups with coefficients in M.

The theorem comes in two parts.

First, it states that there is a natural short exact sequence:

0 \to H^n(X;\mathbb{Z}) \otimes M \to H^n(X;M) \to \operatorname{Tor}(H^{n+1}(X;\mathbb{Z}),M) \to 0

Second, it states that the short exact sequence splits (non-canonically):

H^n(X;M) \cong (H^n(X;\mathbb{Z}) \otimes M) \oplus \operatorname{Tor}(H^{n+1}(X;\mathbb{Z}),M)

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