Basic properties of Neron models

Question

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.)

Answer 1

1) Yes, that is part of Neron's theorem. (There also exist Neron models for semiabelian varieties, which are not of finite type in general.)

2) Yes, for the reason you give ($A$ is connected). When people say "connected component of the Neron model" they generally mean the open subgroup scheme which is in every fibre the connected component of the identity.

An alternative to BLR is the article "Neron models" by M. Artin in "Arithmetic Geometry" (ed. Cornell-Silverman). What you are asking is explained in the first section of that reference (look at 1.2 and 1.16).