Group scheme over a DVR

Question

Let $G$ be a group scheme over a discrete valuation ring $R$, with reductive generic fiber. Would it be the same to require $G$ to have connected generic fiber or to have connected special fiber? Does any of the two conditions imply the other?

Answer 1

If $G$ has connected special fiber, it need not have connected generic fiber: for example, consider the group scheme $\mu_2$ over $\mathbb Z_2$. The generic fiber is isomorphic to $\mathbb Z/2\mathbb Z$, and the special fiber is connected but nonreduced. However, this lemma from the Stacks project says that this behavior can't happen without introducing nilpotents.

In the other direction: Proposition 3.1.12 in Conrad's notes, and its generalization, Theorem 1.2 in Prasad-Yu, give positive answers under some hypotheses. I don't know how necessary these hypotheses are.

Note that if you were interested in more general (non-reductive) group schemes, there are interesting examples with connected generic fiber and disconnected special fibers. For example, Néron models of elliptic curves can have special fibers consisting of several copies of $\mathbb G_m$.