text: An alternative point-of-view can be based on representing cohomology via Eilenberg–MacLane space where the map  takes a homotopy class of maps from  to  to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.
  Example: mod 2 cohomology of the real projective space
  Let , the real projective space. We compute the singular cohomology of  with coefficients in .
  Knowing that the integer homology is given by:
  We have , so that the above exact sequences yield
  In fact the total cohomology ring structure is
  Corollaries
  A special case of the theorem is computing integral cohomology. For a finite CW complex ,  is finitely generated, and so we have the following decomposition.
  where  are the Betti numbers of  and  is the torsion part of . One may check that
  and
  This gives the following statement for integral cohomology:
  For  an orientable, closed, and connected -manifold, this corollary coupled with Poincaré duality gives that .
  Universal coefficient spectral sequence
  There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients.
  For cohomology we have
  Where  is a ring with unit,  is a chain complex of free modules over ,  is any -bimodule for some ring with a unit ,  is the Ext group. The differential  has degree .
  Similarly for homology
  for Tor the Tor group and the differential  having degree .