Let $R$ be a dvr with residue field $k$ and quotient field $K$. Define $S=Spec(R)$. Let $A/K$ be an abelian variety. To my knowledge the Neron model of $A$ is a group scheme ${\cal N}/S$ with generic fibre $A$, which represents the functor $$Y\mapsto Mor_K(Y\times_S Spec(K), A)$$ on the category of smooth $S$-schemes. The morphism ${\cal N}\to S$ is smooth, in particular it is flat and locally of finite type.
Question 1. Is it true that ${\cal N}\to S$ is of finite type?
I know that the special fibre ${\cal N}\times_S Spec(k)$ is in general not connected and that the component group of the special fibre is an important invariant.
But what about the scheme ${\cal N}$ itself?
Question 2. Is it true that ${\cal N}$ is connected?
I strongly assumed that the answer to question 2 is "yes". (My reason to believe this: Assume ${\cal N}$ is not connected. Then ${\cal N}=U\cup V$ for nonempty disjoint open subsets $U$ and $V$ of ${\cal N}$. Then $A=U_K\cup V_K$ and $U_K$, $V_K$ are disjoint open subsets of $A$. Furthermore $U\to S$, $V\to S$ are flat morphisms, hence $U_K$ and $V_K$ are non-empty. This is a contradiction, because $A$ is connected.)
However I saw in the book of Bosch-Lutkebohmert-Raynaud examples of non-connected Neron models. And I saw in Deligne's articles on Hodge theory the expression "connected Neron model of $A$" (as opposed to "Neron model of $A$"). Hence I am very puzzled...
(I have to admit that I did not yet go through the construction of a representing object of the functor above. That is probably the reason why I cannot help myself at the moment.)
