TITLE: Galois cohomology of linear groups over local fields
QUESTION [5 upvotes]: Let $F$ be a local field of characteristic zero (for simplicity), $\overline{F}$ an algebraic closure of $F$ and $L/F$ a fixed finite Galois extension. If $G$ is a linear algebraic group defined over $F$, then the Galois cohomology group $H^1(F,G)$ can be defined as a direct limit of $H^1(K/F,G)$, where $K$ runs through finite Galois subextensions of $\overline{F}$.
Now the question is: under what conditions is this direct limit just $H^1(L/F,G)$?
I guess there might be restrictions on both $L$ and $G$.

REPLY [4 votes]: @Brian Conrad: The sequence $$1 \rightarrow \mu \rightarrow Z' \times \mathcal{G} \rightarrow G' \rightarrow 1$$ is not exact in general. Namely, the kernel $\nu$ of the map
$Z' \times \mathcal{G} \rightarrow G'$ can be bigger than $\mu$. For example, if $G=G'=GL_n$, then $\mathcal{G}= \mathcal{D}(G')=SL_n$, hence $\mu=1$, while $\nu=\mu_n$, the group of roots of unity of order dividing $n$.
Excuse me for posting this as an answer. As a new user, I am not permitted to post a comment.