**Background**: When proving that the group of ![k](http://latex.mathoverflow.net/png?k)-isogenies ![M=\mathrm{Hom}\sb k(A,B)](http://latex.mathoverflow.net/png?M%3D%5Cmathrm%7BHom%7D%5Fk%28A%2CB%29) 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)](http://latex.mathoverflow.net/png?%5Cmathbb%7BZ%7D%5F%5Cell%5Cotimes%5F%7B%5Cmathbb%7BZ%7D%7D%20M%20%5Cto%20%5Cmathrm%7BHom%7D%5F%7B%5Cmathbb%7BZ%7D%5F%5Cell%7D%28T%5F%5Cell%20A%2CT%5F%5Cell%20B%29) is injective. Since each Tate module is free of finite rank over ![\mathbb{Z}\sb \ell](http://latex.mathoverflow.net/png?%5Cmathbb%7BZ%7D%5F%5Cell), it follows that the localization ![M\sb \ell](http://latex.mathoverflow.net/png?M%5F%5Cell) is ![\mathbb{Z}\sb \ell](http://latex.mathoverflow.net/png?%5Cmathbb%7BZ%7D%5F%5Cell)-finite. One then uses a little trick to deduce the ![\mathbb{Z}](http://latex.mathoverflow.net/png?%5Cmathbb%7BZ%7D)-finiteness of ![M](http://latex.mathoverflow.net/png?M) itself. (See Silverman I, for example.)

The above proof needs only a single prime ![\ell](http://latex.mathoverflow.net/png?%5Cell).
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}](http://latex.mathoverflow.net/png?%5Cmathbb%7BZ%7D)-finiteness of ![M](http://latex.mathoverflow.net/png?M) be deduced directly from the ![\mathbb{Z}\sb \ell](http://latex.mathoverflow.net/png?%5Cmathbb%7BZ%7D%5F%5Cell)-finiteness of  ![M\sb \ell](http://latex.mathoverflow.net/png?M%5F%5Cell) for all primes ![\ell](http://latex.mathoverflow.net/png?%5Cell)?

One can consider this a question about general *torsion-free* abelian groups ![M](http://latex.mathoverflow.net/png?M). A non-counterexample to keep in mind is ![M=\mathbb{Z}[1/p]](http://latex.mathoverflow.net/png?M%3D%5Cmathbb%7BZ%7D%5B1%2Fp%5D),
for which ![M\sb \ell](http://latex.mathoverflow.net/png?M%5F%5Cell) is ![\mathbb{Z}\sb \ell](http://latex.mathoverflow.net/png?%5Cmathbb%7BZ%7D%5F%5Cell)-finite for all ![\ell\neq p](http://latex.mathoverflow.net/png?%5Cell%5Cneq%20p).

(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...)