Cochains and the universal coefficient theorem

Let R be a principal ideal domain, let $C_*$ be a chain complex of free R-modules, and let G be an R-module. Define $C^n=\operatorname{Hom}_R(C_n,G)$ and $\delta\varphi=\varphi\circ\partial_{n+1}$. Thus coboundary raises degree, and $\delta^2=0$ follows from boundary squared zero. Cocycles vanish on boundaries; coboundaries evaluate to zero on cycles. Consequently evaluation gives a homomorphism from cohomology to R-linear functionals on homology. The chosen convention has no extra degree sign in the coboundary definition.

If the chain groups have finite chosen bases, this coboundary is represented by the transpose boundary matrix with coefficients acting on G. Infinite bases require care: Hom of a direct sum is a product, so singular cochains can assign nonzero values to infinitely many generating simplices. They are not restricted to finite-support assignments. A continuous map induces a chain map forward and a cochain map backward by precomposition, making cohomology contravariant in spaces, in contrast to homology's covariance.

The universal coefficient theorem gives a short exact sequence with $\operatorname{Ext}^1_R(H_{n-1}(C),G)$ on the left and $\operatorname{Hom}_R(H_n(C),G)$ on the right. The evaluation map is surjective, but the Ext term can represent cohomology classes invisible to evaluation on n-dimensional cycles. It measures the failure of the Hom functor to preserve exactness when applied to the relevant chain relations. The degree n−1 is essential: torsion in one homology degree can contribute to cohomology one degree higher.

The sequence is natural in the appropriate maps of complexes and coefficient modules. Under these PID and free-chain hypotheses it splits, but generally not naturally. One may use a chosen splitting to identify abstract module types, yet cannot silently transport that decomposition through every map of spaces. Over a field the Ext term vanishes and evaluation canonically identifies cohomology with the vector-space dual of homology. Even then an infinite-dimensional dual is not automatically identifiable with the original vector space by a preferred basis.

For integer coefficients, use the free resolution $0\to\mathbb Z\xrightarrow{\times m}\mathbb Z\to\mathbb Z/m\to0$ with positive m. Applying Hom into G shows that $\operatorname{Hom}(\mathbb Z/m,G)$ is the subgroup killed by m, while $\operatorname{Ext}^1(\mathbb Z/m,G)=G/mG$. These two constructions are a kernel and a cokernel of the same multiplication map, not interchangeable definitions. For G equal to Z, the Hom term is zero and the Ext term is Z/m.

For spaces, singular chains satisfy the free-chain hypotheses, and cellular cochains can be used for computations on CW complexes. Degree zero also illustrates the distinction between sums and products: H₀ is the direct sum of one copy of the coefficient ring per path component, while H⁰ with module coefficients gives the corresponding product of assignments. In a finite connected example both reduce to one copy, making it easy to miss this difference until an infinite-component space or an infinite chain basis is considered.