Is quotient of group scheme is again group scheme?

Question

Is quotient of group scheme is again group scheme ?

The following is the case I'm interested in.

Let $K$ be a local field, and $R$ be ring of integers of $K$, then,let ε be a group scheme over Spec$R$ and $ε_0$ be connected component including identity of a group.Then, $ε(k)/ε_0(k)$ is again group scheme ?

Thank you in advance.

Answer 1

Question: "Let $K$ be a local field, and $R$ be ring of integers of $K$, then,let $ε$ be a group scheme over $SpecR$ and $ε_0$ be connected component including identity of a group.Then, $ε(k)/ε_0(k)$ is again group scheme?"

Answer: There are results of the following form: If $k$ is a field and $N⊆G$ is a closed normal subgroup scheme of an affine $k$-group scheme $G$, it follows there is a quotient $G/N$ which is an affine $k$-group scheme. There are generalizations to the case where $k$ is no longer a field.

If you want more specific information you should specify your notation: What is $\epsilon(k)$? Usually $\epsilon(k)$ means the $k$-points of the group scheme $\epsilon$ and the $k$-points of $\epsilon$ is not a scheme in general.

Note: A general reference for group schemes in the more general case: SGA3 Schémas en groupes, 1962–1964 (Group schemes), Lecture Notes in Mathematics 151, 152 and 153, 1970. There are many "elementary" introductory books: Waterhouse "Introduction to affine group schemes" is such a book.