Difficulties in the proof of finiteness of n-Selmer group using cohomology

Question

I was reading the proof of finiteness of n-Selmer group $S^n(E/\mathbb{Q})$ from Milne's Elliptic curve book(1st Edition). While reading the proof I had some difficulties in some arguments.

1st Question: In Milne's book, Lemma 3.8: For any finite Galois extension $L$ of $\mathbb{Q}$ and integer $n \geq 1$, the kernel of $$S^{(n)}(E/\mathbb{Q}) \longrightarrow S^{(n)}(E/L)$$ is finite.

In the proof the author claims that the kernel of $$H^1(\mathbb{Q}, E[n]) \longrightarrow H^1(L, E[n])$$ is finite and the kernel of the map is $H^1(Gal(L/\mathbb{Q}), E(L)[n])$. Where $H^1(K, E)= H^1(Gal(\bar{K}/K),E)$.
I don't understand why the kernel is this.

2nd Question: The author conside the homomorphism $$f : L^\times {\xrightarrow{a \mapsto (ord_{\mathfrak{p}}(a))} }\bigoplus_{\mathfrak{p}\subset \mathcal{O}_L, ~ \mathfrak{p} \text{ prime }} \mathbb{Z},$$ and claimed that cokernel of this map is the ideal class group $C$. My question is why this definition of ideal class group matches with our actual definition using fractional ideal.

3rd Question: corollary 3.12 says that When $T$ is a finite set of prime ideals in $L$, the groups $U_T$ and $C_T$ defined by the exactness of the sequence $$0 \longrightarrow U_T \longrightarrow L^\times {\xrightarrow{a \mapsto (ord_{\mathfrak{p}}(a))} \longrightarrow } \bigoplus_{\mathfrak{p}\notin T} \mathbb{Z} \longrightarrow C_T \longrightarrow 0$$ are, respectively, finitely generated and finite.

I don't understand the meaning of the natation used $U_T$ and $C_T$.

4th Question: How this corollary is useful to prove the next lemma given in the book i.e lemma 3.13 : Assume that $L$ contains the $n^{th}$-unity root. For any finite subset $T$ of $M_L$ containing $M_K^\infty$ , let $N$ be the kernel of $$a \longrightarrow (ord_{\frak{p}}(a) (\mod n)) : L^\times/L^{\times n} \longrightarrow \bigoplus_{\frak{p}\notin T} \mathbb{Z}/n\mathbb{Z}$$ Then there is an exact sequence $$0 \longrightarrow U_T /U_T^n \longrightarrow N \longrightarrow C_T[n]$$ Therefore $N$ is a finite group.

Proof of the lemma 3.13 isn't properly clear to me.

Any kind of help or comment is welcomed.

Thanks in advance.

Answer 1

(Not sure any of these questions are at the right level for this forum, but here the comments that may help.)

1. question : Inflation-restriction sequence.

2. question : The target can be identified with the group of fractional ideals. Each such ideal is a product of prime ideals of the form $\mathfrak{p}_1^{a} \cdot \mathfrak{p}_2^{b} \cdots $ with $a,b\in\mathbb{Z}$; it is a usual ideal if all exponents are positive. Now map this to the vector $(a,b,\dots)$ in the target group. The image of $f$ corresponds to principal ideals. So the cokernel is by definition the class group.

3. I don't understand that question. $U_T$ is defined to be the kernel of the given map $g\colon L^\times \to \bigoplus_{\mathfrak{p}\not\in T} \mathbb{Z}$ and $C_T$ is the cokernel. $U_T$ is the group of $T$-units and Dirichlet's theorem for the usual units also work for $U_T$ to show that it is finitely generated. Similar for $C_T$ and the usual class group.

4. question : Make a diagram with the given exact sequence from the 3rd question having four non-zero terms on top and another copy at the bottom. The vertical maps are multiplication-by-$n$ maps (or powering-by-$n$ maps if the group is written multiplicatively). Now the cokernel of the first vertical map is $U_T/U_T^n$, the cokernel of the second is $L^\times/(L^\times)^n$ and the third vertical map has cokernel $\bigoplus_{\mathfrak{p}\not \in T} \mathbb{Z}/n\mathbb{Z}$. To show the new exact sequence for $N$ is now a diagram chase problem, that I think I should not spell out.