# Faithful representations of integral models

I am reposting a question that I had asked on stackexachage three weeks ago.

Let $$G/\mathbb{Q}$$ be a connected reductive group, and $$\mathcal{G}/\mathbb{Z}$$ be an integral model (i.e. flat affine group scheme of finite type). I was wondering whether there exists a closed immersion $$\mathcal{G}\hookrightarrow \operatorname{GL}_n$$ over $$\mathbb{Z}$$.

I had this question after seeing Excercise 2.1.1 in [Caraiani] which asks the reader to show that any finite index subgroup $$\Gamma\subset\mathcal{G}(\mathbb{Z})$$ is an arithmetic subgroup of $$G(\mathbb{Q})$$ (an arithmetic subgroup of $$G(\mathbb{Q})$$ is one such that $$\Gamma\cap G(\mathbb{Q})\cap \operatorname{GL}_n(\mathbb{Z})$$ has finite index in both $$\Gamma$$ and $$G(\mathbb{Q})\cap \operatorname{GL}_n(\mathbb{Z})$$ for some closed immersion $$G\hookrightarrow \operatorname{GL}_n$$).

My instinct is to believe that my question and the exercise refered to above are the same. This is because, if we can show the exercise, then using some conjugation automorphism of $$\operatorname{GL}_n$$, the closed immersion $$G\hookrightarrow \operatorname{GL}_n$$ should be such that $$\mathcal{G}(\mathbb{Z})$$ maps to $$\operatorname{GL}_n(\mathbb{Z})$$.

I tried to prove this in the same way as one would prove that affine algebriac groups are linear (i.e. have a finite faithful representation). But, the obvious hurdle of working over a ring rather than a field is that modules need not have a basis (more specifically $$H^0(\mathcal{G},\mathcal{O})$$ does not have a basis - even though it is flat over $$\mathbb{Z}$$). I believe the problem can be fixed even if we show that for any $$m\in H^0(\mathcal{G},\mathcal{O})$$, the regular representation $$\mathcal{G}\rightarrow \operatorname{GL}(H^0(\mathcal{G},\mathcal{O}))$$ has sub subrepresentation $$V\subset H^0(\mathcal{G},\mathcal{O})$$ such that $$V$$ is free $$\mathbb{Z}$$-module and $$m\in V$$.

Does someone know if my intuition is correct? Can we get such a closed immersion? Or are the two problems discussed above not equivalent?

Reference:

Caraiani, Ana, Lecture notes on Perfectoid Shimura varieties.

Yes, there exists a closed immersion $$\mathcal{G}\to \mathrm{GL}_n$$ over $$\mathbb{Z}$$. This is folklore. For a proof, see for example Proposition 3 of arXiv:2012.05708v3