Question

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.

Answer 1

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.

Answer 2

They are using continuous cohomology, so that $$ H^n(G,M) = \varinjlim H^n(G/H,M^H) $$ if $G$ is topological and $M$ is discrete (thanks Arjit) $G$-module (the limit runs over open compact subgroups $H$ of $G$). Look in p. 38 for the definition.