Special and general types Special notions Variants Extra structure Operations Theorems The universal coefficient theorem states how ordinary homology/ordinary cohomology determines homology/cohomology with arbitrary coefficients. For a chain complex (of abelian groups) and a field (the coefficient field), the homology group and the cohomology group are indeed related by dualization: . If the coefficients are not a field but an arbitrary abelian group, then this relationship receives a correction by an Ext-group. This is discussed below in For ordinary cohomology. Dually, again if is a field then there is an isomorphism and for more general this is corrected by a Tor-group. This is discussed below in For ordinary homology. More generally, under suitable conditions there are universal coefficient theorems that relate generalized (Eilenberg-Steenrod) cohomology to the dual of generalized homology. This is discussed below in For generalized cohomology. There is also a version of the theorem for Kasparov’s KK-theory, see the references. Let be a chain complex of free abelian groups. Let be an arbitrary abelian group. Write for the dual cochain complex with respect to ; for the chain homology of for the cochain cohomology of hence for the cochain cohomology of with coefficients in . There is a canonical morphism of abelian groups given by sending a cocycle to evaluation of that cocycle on a chain: (universal coefficient theorem in ordinary cohomology) The morphism is surjective and its kernel is the Ext group . In other words, there is a short exact sequence hence Moreover, this sequence splits (non-canonically). We reproduce the direct proof given for instance in (Boardman).