P-adic representations - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:19:27Z http://mathoverflow.net/feeds/question/80637 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80637/p-adic-representations P-adic representations Arijit 2011-11-10T21:04:54Z 2011-11-25T10:11:27Z <p>Hi, I am reading about p-adic representations from Fontaine's book which can be found at <a href="http://staff.ustc.edu.cn/~yiouyang/research.html" rel="nofollow">http://staff.ustc.edu.cn/~yiouyang/research.html</a>. 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> http://mathoverflow.net/questions/80637/p-adic-representations/80647#80647 Answer by unknown for P-adic representations unknown 2011-11-11T00:41:25Z 2011-11-11T01:11:08Z <p>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 (<em>thanks Arjit</em>) $G$-module (the limit runs over open compact subgroups $H$ of $G$). Look in p. 38 for the definition.</p> http://mathoverflow.net/questions/80637/p-adic-representations/81861#81861 Answer by Laurent Berger for P-adic representations Laurent Berger 2011-11-25T10:11:27Z 2011-11-25T10:11:27Z <p>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". </p> <p>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.</p>