Editing Universal coefficient theorem (section)

==Corollaries==
A special case of the theorem is computing integral cohomology. For a finite [[CW complex]] <math>X</math>, <math>H_i(X,\Z)</math> is finitely generated, and so we have the following [[Fundamental theorem of finitely generated abelian groups#Classification|decomposition]].

:<math> H_i(X; \Z) \cong \Z^{\beta_i(X)}\oplus T_{i},</math>

where <math>\beta_i(X)</math> are the [[Betti number]]s of <math>X</math> and <math>T_i</math> is the torsion part of <math>H_i</math>. One may check that

:<math> \operatorname{Hom}(H_i(X),\Z) \cong \operatorname{Hom}(\Z^{\beta_i(X)},\Z) \oplus \operatorname{Hom}(T_i, \Z) \cong \Z^{\beta_i(X)},</math>

and

:<math>\operatorname{Ext}(H_i(X),\Z) \cong \operatorname{Ext}(\Z^{\beta_i(X)},\Z) \oplus \operatorname{Ext}(T_i, \Z) \cong T_i.</math>

This gives the following statement for integral cohomology:

:<math> H^i(X;\Z) \cong \Z^{\beta_i(X)} \oplus T_{i-1}. </math>

For <math>X</math> an [[orientability|orientable]], [[closed manifold|closed]], and [[connected space|connected]] <math>n</math>-[[manifold]], this corollary coupled with [[Poincaré duality]] gives that <math>\beta_i(X)=\beta_{n-i}(X)</math>.