Let $k$ be a field and let $\mathbf{AV}_{/k}$ be the category of abelian varieties over $k$. I'm interested in different definitions of the isogeny category of $\mathbf{AV}_{/k}$. Of course, the isogeny category will have the same objects, so it's only about the Hom-sets, which are denoted by $\mathrm{Hom}^0(A,B)$. The easiest definition of the isogeny category is probably $\mathrm{Hom}^0(A,B):=\mathbb Q\otimes_{\mathbb Z} \mathrm{Hom}(A,B)$. I understand that this is equivalent to saying $\mathrm{Hom}^0(A,B)=\varinjlim\limits_{A' \to A}\mathrm{Hom}(A',B)$ where the colimit is taken over all isogenies $A' \to A$.
Morally, the isogeny category should be some kind of localization at isogenies. So one might try to simply let $S$ be the class of isogenies and then ask if $S^{-1}\mathbf{AV}_k$ is equivalent to the isogeny category as defined above. This feels like it should be true, but I don't see a formal proof.
Furthermore, I thought up a different way to define potentially the same category: we can embed $\mathbf{AV}_{/k}$ into the category of all commutative group schemes of finite type over $k$, which is an abelian category. The category of finite group schemes is a Serre subcategory, so one can form the Serre localization. We can restrict the localization functor to $\mathbf{AV}_{/k}$ and consider the essential image. Again, it feels like this should be equivalent to the isogeny category, but I don't see a formal proof. (Incidentally this yields something like the isogeny category for commutative group schemes as a by-product which might interesting in itself.)
I'm asking for an explanation why these three approaches to the isogeny category are equivalent.
