Let $X, Y$ be abelian varieties over $k$. Let $l$ be a prime not equal to the characteristic of $k$. Then one shows that $\text{Hom}_k(X, Y)\to \text{Hom}_{\mathbb{Z}_l}(T_l X, T_l Y)$ is injective. Moreover, it is shown that the $\mathbb{Z}_l$ linear map $\text{Hom}_k(X, Y)\otimes_{\mathbb{Z}} \mathbb{Z}_l \to \text{Hom}_{\mathbb{Z}_l}(T_l X, T_l Y), f\otimes c\mapsto c\cdot T_l(f)$ is injective with torsion-free cokernel and that $T_l X$ and $T_l Y$ are non-canonically isomorphic to free $\mathbb{Z}_l$ modules of finite rank.
Then it is concluded that $\text{Hom}_k(X,Y)$ is a free abelian group of finite rank. I can not understand how that can be concluded from the above. (See for example Ben Moonen's draft for a book on abelian varieties chapter 15).
At first, this seemed rather obvious but check out the second answer to the question Checking that a torsion-free abelian group has finite rank here on math-exchange.