Hi! Let a finitely generated free abelian group and a homomorphism, then one can define a trace as usually as sum of the diagonal elements (of the matrix form of ).

If is not free, then is free ( denotes the torsion group) and one has an induced homomorphism . Therefore we can define the trace of as .

Now my question:

Let finitely generated abelian groups and

a commuative diagramm with exact rows (there should be vertical arrows: , but i do not know how to produce this. I hope it is clear what i mean).

Why do we have in this case?

My problem is that the induced sequence between the quotients is not longer exact anymore, since we lose the property .

Does somebody have a clue how to prove this?