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I am reading Massey's book on algebraic topology and on the chapter of universal coefficient theorem of cohomology, there is this exercise 4.1 that I don't know how to solve.

Let (X,A)be a pair such that $H_n(X,A)$ is a finitely generated abelian group.

Prove rank$(H^n(X,A;\mathbb{Z})) = \text{rank}(H_n(X,A))$,Torsion$(H^n(X,A;\mathbb{Z})) \cong \text{Torsion }(H_{n-1}(X,A))$.

I wonder is this a corollary of the universal coefficient thereom? If so, how do we compute Tor(Ext($H_n(X,A)$))?

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The fact that the torsion "shifts by one" in cohomology and the free part (therefore the rank) stays isomorphic, is basically one of the first things which you learn in cohomology. Compare for instance Corollary 3.3 from Hatcher's Algebraic Topology, which answers precisely your question. Computations are provided above this corollary.

The link to the book:

http://www.math.cornell.edu/~hatcher/AT/AT.pdf

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