TITLE: GIT over integers
QUESTION [11 upvotes]: Let $G$ be a reductive algebraic group over ${\mathbb Z}$ (or a finite
localization of a ring of integers $R$ in a number field) acting on an
affine scheme of finite type $M=Spec(A)$ over $R$. 
We can take the invariants $A^G$, and then reduce
modulo a prime $p$: $A^G\otimes_{\mathbb Z}{\mathbb F}_q$. We can first
reduce modulo $p$, and then take invariants: 
$(A\otimes_{\mathbb Z}{\mathbb F}_q)^{G_p}$. Is it true that for almost all
primes $p$ the results coincide?
Let $M^\theta\subset M$ be the open subset of $\theta$-stable
points for a choice of a character $\theta$ of $G$. 
Let $M^{\theta,f}\subset M^\theta$ be the open subset of $M^\theta$ formed by
all the points with trivial stabilizers. Let $M//_\theta G$ be the GIT quotient
(the projective spectrum of the ring of $\theta$-semiinvariants of the 
$G$-action on $A$). Is it true that the natural projection
$M^{\theta,f}\to M//_\theta G$ is a $G$-torsor? (\'etale locally trivial?)

REPLY [5 votes]: The answer to the first question is positive. One does not need normality of the 
generic fiber. By the universal coefficient theorem [Prop I 4.18 (a) in Jantzen 
Representations of algebraic groups], it suffices to know that $H^1(G,A)$ vanishes
generically on $Spec(R)$. But this is implied by theorem 33 of my paper with Franjou 
Power reductivity over an arbitrary base.
Using further suggestions by user52824 we see that the right setting has a Dedekind domain $D$ as base ring. We localize by inverting a nonzero element of $D$. 
We first localize to get $A$ flat over $D$, then we localize further to make
$H^1(G,A)$ flat over $D$. This uses Theorem 10.5 of Good Grosshans filtration in a family as the big hammer. Then the universal coefficient theorem implies compatibility with any base change. Notice that we no longer try to kill $H^1(G,A)$. Just its $D$-torsion.