I am currently reading https://arxiv.org/abs/math/0703310 and I was wondering why the map $S \to B_SG''$ in proof of proposition 2.7 (c) is faithfully flat.
This is as far as I already understood things:
As they are aiming to apply Proposition 2.6 (b), it really should be thought about the morphism $S \to G''$ as group schemes and flatness & surjectivity should be proven for this morphism (here $G''$ is a group scheme over $S$).
As a general check of validity, this statement should be true for affine group schemes in particular.
Therefore, assume that $S = Spec R$ and $G'' = Spec A''$ where $A''$ is a Hopf algebra over $R$. Then the map corresponding to $S \to G''$ should be $A'' \to R$ or in other words: The counit map!
To show that a map of affine schemes is surjective is equivalent to showing that the map of Hopf algebras is injective.
However, it was shown in another post here, that the counit map is only injective if the Hopf algebra is trivial.
What exactly is happening here and where is my mistake?