# split integral model of a reductive group

Let $$F$$ be a number field, $$p\in\mathbb{Z}$$ a prime which is unramified in $$F$$ and $$G$$ a connected reductive group over $$F$$. Moreover $$G$$ is supposed to be quasi-split over $$p$$.

Does there exist a finite set of primes $$S\subset\mathbb{Z}$$ which does not contain $$p$$ and a split reductive group $$\mathcal{G}$$ defined over $$\mathbb{Z}[1/S]$$ such that for some finite extension $$K/F$$ which is not ramified over $$p$$ we have the following isomorphism:

$$\mathcal{G}\times_{\mathbb{Z}[1/S]}K\cong G\times_FK.$$

More generally, given a reductive group $$G/F$$, does there exist some sort of split integral model $$\mathcal{G}$$ defined over rings like $$\mathbb{Z}$$, $$\mathbb{Z}[1/S]$$ or $$\mathcal{O}_F$$?

• Thanks for your answer. What does it mean by saying that $G$ is unramified over $p$? I am sorry, I am not quite familiar with some terminologies in algebraic group. – tanjia Sep 21 '19 at 3:36