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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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