1. 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?
2. 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?)