TITLE: Torsors over complete local fields
QUESTION [6 upvotes]: Let $G$ be a linear algebraic group scheme, and let $R$ be a complete discrete valuation ring, with quotient field $K$ and residue field $k$.
If $T$ is an $R$-torsor, it yields by base change a $k$-torsor $T_k$.
Apparently, we have the following theorem:
Thm. If $G$ is smooth and $T,T'$ are two $R$-torsors, then $T_k\simeq T'_k$ implies that $T\simeq T'$.
This is proved in SGA3,p. 401,prop. 8.1. In fact, more is proved in SGA3 (it says that any $k$-torsor can be lifted in a unique way to an $R$-torsor, up to isomorphism), but I only need this particular case.
Unfortunately, I do not have access to SGA3, and I'm not really comfortable with the different topologies, so here is a couple of questions:

Is the topology used in the theorem the étale topology ?
Would it be true if $G$ is not necessarily smooth but if we use torsors for the fppf topology ?

Typically, I wonder what would happen for $G=\mu_2$ defined over a field of characteristic $2$ ? 
Thanks a lot!

REPLY [6 votes]: The isomorphism classes of torsors for $\mu_{n}$ over $R$ and $k$ are
classified by the fppf cohomology groups $H^{1}(R,\mu_{n})$ and $H^{1}%
(k,\mu_{n})$. From the Kummer sequence, one sees that $H^{1}(R,\mu_{n})\simeq
R^{\times}/R^{\times n}$ and $H^{1}(k,\mu_{n})\simeq k^{\times}/k^{\times n}$.
If $n$ is prime to the residue characteristic (so $\mu_{n}$ is etale),
then $R^{\times}/R^{\times n}\rightarrow k^{\times}/k^{\times n}$ is a
bijective, but if $n$ equals the residue characteristic then it is not injective;
this means that there are nonisomorphic torsors over $R$ that become
isomorphic over $k$. (Yes, a key point is that for smooth groups, the torsors are classified by the etale cohomology groups.)