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Let $R = \mathbb{Z} [ \frac{1}{p}]$ for some prime number $p$ and $GL_{n,R}$ be the general linear group scheme over $R$. The bar construction gives a simplicial scheme $BGL_{n,R}$ over the constant simplicial scheme $Spec(R)$. If $q$ is a prime different from $p$ we can pull $BGL_{n,R}$ back along a map $Spec( \bar{\mathbb{F}_q}) \to Spec(R)$ to get $BGL_{n,\bar{\mathbb{F}_q}}$. Here $\bar{\mathbb{F}_q}$ is an algebraic closure of $\mathbb{F}_q$. The simplicial scheme $BGL_{n,\bar{\mathbb{F}_q}}$ has the nice property that if we apply Friedlander's étale topological type functor, defined here, and then p-complete, we get something that is equivalent to the $p$-completion tower $ \{ (\mathbb{Z}/p)_s BGL_n( \mathbb{C}) \}_s $. (Here $BGL_{n}( \mathbb{C})$ means the singular simplicial set of the classifying space of the Lie group).

Several articles state that the sequence$$(BGL_{n,\bar{\mathbb{F}_q}})_{ét} \to (BGL_{n,R})_{ét} \to Spec(R)_{ét}$$ becomes a fibration sequence after $p$-completing the $BGL$ terms, but I haven't been able to find any proof or argument supporting this anywhere. Does anyone know of a proof or argument for this?

In the article Exotic cohomology for $GL_n(\mathbb{Z} [ \frac{1}{2}])$ the reader is referred to Étale homotopy of simplicial schemes but I have only been able to find a proof of the $p$-adic equivalence I mentioned above, not of the fibration sequence. In Algebraic and étale k-theory it is used several times.

I hope this question isn't too narrow for Mathoverflow.

The reason that I ask is that I would like to have similar fibration sequences for other group schemes and I hope they will be fibration sequences for the same reason that the one above is.

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There are at least two ways this could be done, maybe more.

The easier way would be to write the ind-scheme $BGL_n$ over $\operatorname{Spec}R$ as colimit of the Grassmannians $Gr(n,k)$ for $k\to\infty$. Because the Grassmannians are proper and smooth you can use Corollary 4.8 of

  • E.M. Friedlander. Fibrations in étale homotopy theory. Publ. Math. IHES 42 (1973), 5-46.

This provides fiber sequences $(Gr(n,k)_{\overline{\mathbb{F}_q}})_{ét}\to(Gr(n,k)_{R})_{ét}\to(\operatorname{Spec} R)_{ét}$ . Then there is a passage to a colimit, which should be no problem because it's a directed colimit (contractible index category).

However, the above does not seem to work for arbitrary group schemes because it needs smooth proper approximations of classifying spaces. The other way, which could possibly work more generally would be to follow the argument for the proof of Corollary 4.8 and Theorem 5.3 in the abovementioned paper for $BGL_n$ directly. Essentially, you need to prove the following: for $f:(BGL_n)_{ét}\to(\operatorname{Spec} R)_{ét}$ and $A$ a pro-$p$ abelian group, $R^qf_\ast A$ is locally constant and has fiber $H^\ast_{ét}(BGL_n,A)$ for each geometric point. This seems very plausible; could be that this follows from smooth base change plus the p-completion equivalence mentioned in the question but I haven't checked the details.

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