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I'm reading and trying to understand the proof of the finiteness of n-Selmer group from J.S.Milne's Elliptic Curves book but having difficulty in understanding it. Here's a screenshot from the book- enter image description here

Now, what I don't understand in this proof is- since we want to show the exactness of a sequence it suffices to prove it at $N$, but I don't understand how the proof in the book implies it.

Also, how can I make sense of the map $[n]$ from $C_{T}$ to $C_{T}$?

I would appreciate if it if someone could dumb it down (the proof) or maybe could suggest some reference where the intermediate steps are done.

Thank you!

EDIT: Also, I would like it very much if someone could suggest some references where I can the theory related to the finite extensions of $L_v$, where $L$ is a number field and $L_v$ is it's completion with respect to a valuation. It's making me very frustrated unable to fill the gaps in the proof because I don't have the knowledge of some pre requisite algebra. I'd appreciate any help.

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  • $\begingroup$ About the edit, the standard reference is the book Local fields, by J.P. Serre (if I understand what you are asking for). For the previous question, if $C_T$ is an abelian group, $[n]$ is just multiplication by $n$. $\endgroup$
    – Xarles
    Jul 9, 2019 at 21:05
  • $\begingroup$ For the proof, it is a combination of two applications of the "snake lemma". You can divide the fourth step exact sequences into two ahort exact sequences. $\endgroup$
    – Xarles
    Jul 9, 2019 at 21:16

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