show/hide this revision's text 2 changed title (finite-->finitely generated)

Is a torsion free abelian group finitefinitely generated, if all of its localizations at primes p are finite finitely generated over Zp?
show/hide this revision's text 1

Is a torsion free abelian group finite, if all of its localizations at primes p are finite over Zp?

Background: When proving that the group of k-isogenies M=\mathrm{Hom}\sb k(A,B) between two abelian varieties is finitely generated, one first shows that the Tate map \mathbb{Z}\sb \ell\otimes\sb {\mathbb{Z}} M \to \mathrm{Hom}\sb {\mathbb{Z}\sb \ell}(T\sb \ell A,T\sb \ell B) is injective. Since each Tate module is free of finite rank over \mathbb{Z}\sb \ell, it follows that the localization M\sb \ell is \mathbb{Z}\sb \ell-finite. One then uses a little trick to deduce the \mathbb{Z}-finiteness of M itself. (See Silverman I, for example.)

The above proof needs only a single prime \ell. But disregarding issues of the characteristic of the field (which are apparently surmountable) we actually have an injective Tate map at every prime. Thus...

Question: Can the \mathbb{Z}-finiteness of M be deduced directly from the \mathbb{Z}\sb \ell-finiteness of M\sb \ell for all primes \ell?

One can consider this a question about general torsion-free abelian groups M. A non-counterexample to keep in mind is M=\mathbb{Z}[1/p], for which M\sb \ell is \mathbb{Z}\sb \ell-finite for all \ell\neq p.

(A google search shows that there is actually quite a body of literature on torsion-free abelian groups, so perhaps the answer to this question is well-known, but I'm not sure where to look...)