Question of algebraic flavor in algebraic topolgy

[quasar987]

See Hatcher p.146-147 theorem 2.44: http://www.math.cornell.edu/~hatcher/AT/ATch2.pdf

In the proof, he puts himself in the more general situation where he has a bounded chain complex of finitely generated groups and goes on to prove that the alternating sum of the ranks of these groups equals the alternating sum of the ranks of their homology groups. From there, the result follows as a special case (in view of lemma 2.34 and theorem 2.35).

The proof relies of the little algebraic fact stated between the statement of the theorem and its proof. I believe that in this little algebraic fact (which is trivial given the fundamental theorem on finitely generated abelian groups), the condition that the groups be finitely generated is important.

What I wonder is, in the proof, why are the cycle groups Z_n=Ker(d_n) finitely generated??

(Note that in the case that interests us for the statement of the theorem this is immediate because the groups in the cellular chain complex are actually finitely generated free abelian and subgroups of finitely generated free abelian groups are themselves finitely generated free abelian...)

 

[yyat]

A subgroup of a finitely generated abelian group is finitely generated:

Let A be a finitely generated abelian group, B a subgroup of A. Then there exists a surjective homomorphism $$f:\mathbb{Z}^n\to A$$ and $$C=f^{-1}(B)$$ is a subgroup of $$\mathbb{Z}^n$$, hence free and finitely generated. A set of generators of C is mapped to a set of generator of B by f, thus B is also finitely generated.

 

[quasar987]

Ahh, very nice! Thanks yyat.