Editing Universal coefficient theorem (section)

==Universal coefficient theorem for cohomology==
Let <math>G</math> be a module over a [[principal ideal]] domain <math>R</math> (for example <math>\Z</math>, or any field.)

There is a '''universal coefficient theorem for [[cohomology]]''' involving the [[Ext functor]], which asserts that there is a natural short exact sequence

:<math> 0 \to \operatorname{Ext}_R^1(H_{i-1}(X; R), G) \to H^i(X; G) \, \overset{h} \to \, \operatorname{Hom}_R(H_i(X; R), G)\to 0.</math>

As in the homology case, the sequence splits, though not naturally. In fact, suppose

:<math>H_i(X;G) = \ker \partial_i \otimes G / \operatorname{im}\partial_{i+1} \otimes G,</math>

and define

:<math>H^*(X; G) = \ker(\operatorname{Hom}(\partial, G)) / \operatorname{im}(\operatorname{Hom}(\partial, G)).</math>

Then <math>h</math> above is the canonical map:

:<math>h([f])([x]) = f(x).</math>

An alternative point of view can be based on representing cohomology via [[Eilenberg–MacLane space]], where the map <math>h</math> takes a [[homotopy]] class of maps <math>X\to K(G,i)</math> to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a ''weak right [[adjoint functor|adjoint]]'' to the homology [[functor]].<ref>{{Harv|Kainen|1971}}</ref>