Let $S$ be a scheme and $X$ a smooth separated faithfully flat over $S$.

An $S$-birational group law on $X$ is an $S$-rational map $$m:X\times_S X\dashrightarrow X, (x,y)\mapsto xy$$ such that

a) the $S$-rational maps $$\Phi:X\times_S X\dashrightarrow X\times_S X, (x,y)\mapsto(x,xy)$$ $$\Psi:X\times_S X\dashrightarrow X\times_S X, (x,y)\mapsto(xy,y)$$ are $S$-birational, and

b) m is associative; i.e., (xy)z=x(yz) whenever both sides are defined.

Now, assume that $S$ is a valuation ring and that the generic and special fibers of $X$ have birational group laws (for instance if they are group schemes).

When and how is it possible to get an $S$-birational group law on $X$?

## Context:

Let $K$ be a valuation field with valuation ring $R$. Let $H$ be a (qc, separated, integral but not necessarily of finite type) group scheme over $R$. Assume that the generic fiber $H_K$ is an algebraic group (i.e. of finite type). We may write $H=\varprojlim_i H_i$ for integral, separated, of finite type $H_i$ over $R$. They may not be group schemes over $R$. But $(H_i)_K$ is an algebraic group. Assume that for large enough $i$, the inverse limit is birational on the special fiber, so for large enough $i$, $(H_i)_k$ has a $k$-birational group law.

Putting the smoothness conditions aside, does $H_i$ have an $R$-birational group law for large enough $i$?

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