Hi,
 I am reading about p-adic representations from Fontaine's book which can be found at <http://staff.ustc.edu.cn/~yiouyang/research.html>. On page 145
where they prove Proposition 5.24 which is essentially the theorem
of Tate-Sen, they show $H^{n}(Gal(L/K\_{\infty},C(i)^{G\_L})=0$ and
the argument is essentially same as in the proof of hilbert Thm 90. But
then they are concluding that implies $H^{n}(H\_K,C(i))=0$ by passing to the
limit. I am confused because I thought that you can only pass through
the limit in case of discrete modules. I think this same argument will
also show that $H^{1}(G\_K,C\_K)=0$ which is not true. I am sure I am missing something obvious. I will greatly appreciate any kind of clarification.



P-adic representations
The whole thing is done with more details in Tate's original article "p-divisible groups", section 3.2. Tate proves that one can approximate a cocyle in $C\_p(i)$ by cocyles with values in $Q\_p^{alg}(i)$ and this is how he reduces the computation to the "discrete case". 


I would suggest that it's better to prove the result by $p$-adic approximation. This way, you can basically work with cocycles with values in $O\_{C\_p}(i)/p^n$, also a discrete space.


