Consider the tensor product of modules ${\displaystyle H_{i}(X,\mathbb {Z} )\otimes A}$ . The theorem states there is a short exact sequence involving the Tor functor

${\displaystyle 0\to H_{i}(X,\mathbb {Z} )\otimes A\,{\overset {\mu }{\to }}\,H_{i}(X,A)\to \operatorname {Tor} _{1}(H_{i-1}(X,\mathbb {Z} ),A)\to 0.}$ 

Furthermore, this sequence splits, though not naturally. Here ${\displaystyle \mu }$  is the map induced by the bilinear map ${\displaystyle H_{i}(X,\mathbb {Z} )\times A\to H_{i}(X,A)}$ .

If the coefficient ring ${\displaystyle A}$  is ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ , this is a special case of the Bockstein spectral sequence.

Let ${\displaystyle G}$  be a module over a principal ideal domain ${\displaystyle R}$  (for example ${\displaystyle \mathbb {Z} }$ , 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

${\displaystyle 0\to \operatorname {Ext} _{R}^{1}(H_{i-1}(X;R),G)\to H^{i}(X;G)\,{\overset {h}{\to }}\,\operatorname {Hom} _{R}(H_{i}(X;R),G)\to 0.}$ 

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

${\displaystyle H_{i}(X;G)=\ker \partial _{i}\otimes G/\operatorname {im} \partial _{i+1}\otimes G,}$ 

and define

${\displaystyle H^{*}(X;G)=\ker(\operatorname {Hom} (\partial ,G))/\operatorname {im} (\operatorname {Hom} (\partial ,G)).}$ 

Then ${\displaystyle h}$  above is the canonical map:

${\displaystyle h([f])([x])=f(x).}$ 

An alternative point of view can be based on representing cohomology via Eilenberg–MacLane space, where the map ${\displaystyle h}$  takes a homotopy class of maps ${\displaystyle X\to K(G,i)}$  to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.^[1]

Let ${\displaystyle X=\mathbb {RP} ^{n}}$ , the real projective space. We compute the singular cohomology of ${\displaystyle X}$  with coefficients in ${\displaystyle G=\mathbb {Z} /2\mathbb {Z} }$  using integral homology, i.e., ${\displaystyle R=\mathbb {Z} }$ .

Knowing that the integer homology is given by:

${\displaystyle H_{i}(X;\mathbb {Z} )={\begin{cases}\mathbb {Z} &i=0{\text{ or }}i=n{\text{ odd,}}\\\mathbb {Z} /2\mathbb {Z} &0<i<n,\ i\ {\text{odd,}}\\0&{\text{otherwise.}}\end{cases}}}$ 

We have ${\displaystyle \operatorname {Ext} (G,G)=G}$  and ${\displaystyle \operatorname {Ext} (R,G)=0}$ , so that the above exact sequences yield

${\displaystyle H^{i}(X;G)=G}$ 

for all ${\displaystyle i=0,\dots ,n}$ . In fact the total cohomology ring structure is

${\displaystyle H^{*}(X;G)=G[w]/\left\langle w^{n+1}\right\rangle .}$ 

A special case of the theorem is computing integral cohomology. For a finite CW complex ${\displaystyle X}$ , ${\displaystyle H_{i}(X,\mathbb {Z} )}$  is finitely generated, and so we have the following decomposition.

${\displaystyle H_{i}(X;\mathbb {Z} )\cong \mathbb {Z} ^{\beta _{i}(X)}\oplus T_{i},}$ 

where ${\displaystyle \beta _{i}(X)}$  are the Betti numbers of ${\displaystyle X}$  and ${\displaystyle T_{i}}$  is the torsion part of ${\displaystyle H_{i}}$ . One may check that

${\displaystyle \operatorname {Hom} (H_{i}(X),\mathbb {Z} )\cong \operatorname {Hom} (\mathbb {Z} ^{\beta _{i}(X)},\mathbb {Z} )\oplus \operatorname {Hom} (T_{i},\mathbb {Z} )\cong \mathbb {Z} ^{\beta _{i}(X)},}$ 

and

${\displaystyle \operatorname {Ext} (H_{i}(X),\mathbb {Z} )\cong \operatorname {Ext} (\mathbb {Z} ^{\beta _{i}(X)},\mathbb {Z} )\oplus \operatorname {Ext} (T_{i},\mathbb {Z} )\cong T_{i}.}$ 

This gives the following statement for integral cohomology:

${\displaystyle H^{i}(X;\mathbb {Z} )\cong \mathbb {Z} ^{\beta _{i}(X)}\oplus T_{i-1}.}$ 

For ${\displaystyle X}$  an orientable, closed, and connected ${\displaystyle n}$ -manifold, this corollary coupled with Poincaré duality gives that ${\displaystyle \beta _{i}(X)=\beta _{n-i}(X)}$ .

There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients.

For cohomology we have

${\displaystyle E_{2}^{p,q}=\operatorname {Ext} _{R}^{q}(H_{p}(C_{*}),G)\Rightarrow H^{p+q}(C_{*};G),}$ 

where ${\displaystyle R}$  is a ring with unit, ${\displaystyle C_{*}}$  is a chain complex of free modules over ${\displaystyle R}$ , ${\displaystyle G}$  is any ${\displaystyle (R,S)}$ -bimodule for some ring with a unit ${\displaystyle S}$ , and ${\displaystyle \operatorname {Ext} }$  is the Ext group. The differential ${\displaystyle d^{r}}$  has degree ${\displaystyle (1-r,r)}$ .

Similarly for homology,

${\displaystyle E_{p,q}^{2}=\operatorname {Tor} _{q}^{R}(H_{p}(C_{*}),G)\Rightarrow H_{*}(C_{*};G),}$ 

for ${\displaystyle \operatorname {Tor} }$  the Tor group and the differential ${\displaystyle d_{r}}$  having degree ${\displaystyle (r-1,-r)}$ .

• Allen Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage.
• Kainen, P. C. (1971). "Weak Adjoint Functors". Mathematische Zeitschrift. 122: 1–9. doi:10.1007/bf01113560. S2CID 122894881.
• Jerome Levine. “Knot Modules. I.” Transactions of the American Mathematical Society 229 (1977): 1–50. https://doi.org/10.2307/1998498

• Universal coefficient theorem with ring coefficients