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For $S$ a noetherian scheme, let $\mathcal{A}(S)$ be the additive category of abelian schemes over $S$ and $\mathcal{A}_{\mathbb{Q}}(S)$ be the category of abelian schemes up to isogenies, i.e. obtained equivalently by formally inverting isogenies or by tensoring the Hom groups with $\mathbb{Q}$.

If $S$ is normal connected with generic point $\eta$, it is known that the functor $\mathcal{A}(S)\rightarrow \mathcal{A}(\eta)$ is fully faithful (see e.g. Chai-Faltings, Degeneration of abelian varieties, I Proposition 2.7), which implies the same for $\mathcal{A}_{\mathbb{Q}}$. Let us say that an object in the essential image of $\mathcal{A}(S)\rightarrow \mathcal{A}(\eta)$ has good reduction. By Poincare irreductibility (which holds over any field), the category $\mathcal{A}_{\mathbb{Q}}(\eta)$ is abelian semi-simple. Of course, this implies that any morphism of abelian schemes can be factored up to isogenies at the generic point into the composition of a projector and an inclusion as direct factor, but having good reduction is not stable by isogeny, and general morphisms of abelian schemes seem (to my novice eye) potentially very complicated.

Suppose now that $S$ either has all residual characteristics equal to 0 or is a Dedekind scheme. Then I can prove that the full subcategory $\mathcal{A}_{\mathbb{Q}}(S)$ of $\mathcal{A}_\mathbb{Q}(\eta)$ is stable by kernels and cokernels, so in particular is abelian. In characteristic 0, this is implied by Grothendieck's theorem which says that to extend an abelian scheme it is enough to extends the l-adic Tate module (Cor 4.2 in the paper "Un th\'eor\'eme sur les homomorphismes des schemas abeliens") plus the fact that the etale fundamental group of a dense open surjects onto the fundamental group of $S$. For Dedekind schemes, this is a consequence of the theory of Neron models, taking the Neron model of the generic kernel/cokernel (and here inverting the residual characteristic is especially important since there are homomorphisms of abelian schemes over a DVR with finite non-flat kernel).

My question is: is this true on a general normal scheme ?

I expect the answer to be no, and tried to construct such bad morphisms out of morphisms of finite flat group schemes with non-flat kernels, but I did not manage to find a counter-example. I am not an arithmetic geometer, and I hope the question is not too naive.

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