Open subgroup of a valuation ring in Lemma 2.3 in Milne's Class Field Theory

Question

I'm reading Milne's Class Field Theory. On the page 104, it gives the following lemma:

In the lemma, $K$ is a local field, and $\pi$ is the uniformizer of $L$.

LEMMA $2.3$ Let $L$ be a finite Galois extension of $K$ with Galois group $G$. Then there exists an open subgroup $V$ of $\mathcal{O}_{L}$, stable under $G$, such that $H^{r}(G, V)=0$ for all $r>0$.

PROOF. Let $\left\{x_{\tau} \mid \tau \in G\right\}$ be a normal basis for $L$ over $K$ (see FT, 5.18). The $x_{\tau}$ have a common denominator $d$ in $\mathcal{O}_{K}$ (see ANT, 2.6). After replacing each $x_{\tau}$ with $d \cdot x_{\tau}$, we may suppose that they lie in $\mathcal{O}_{L} .$ Take $V=\sum \mathcal{O}_{K} x_{\tau} .$ It is stable under $G$ because the normal basis is. Let $\pi$ be a prime element of $\mathcal{O}_{L}$. Then $V \supset \pi^{m} \mathcal{O}_{L}$ for some $m>0$, which shows that $V$ is open (it is union of cosets of $\left.\pi^{m} \mathcal{O}_{L}\right) .$ Finally, $$ V \simeq \mathcal{O}_{K}[G] \simeq \operatorname{Ind}^{G} \mathcal{O}_{K} $$ as $G$ -modules, and so $H^{r}(G, V)=0$ for all $r>0(\mathrm{II}, 1.12)$.

I have several questions:

1. Since elements in $Gal(L/K)$ preserves the valuation of each element, we know that elements in the normal basis have the same valuation. But it seems that elements in $\bigoplus_{\sigma \in G} x_{\sigma}K$ can not contain all elements $\pi^{\mathbb Z}$. Because the difference on the valuation of two elements in $\bigoplus_{\sigma \in G} x_{\sigma}K$ is at least $e$(the ramification index). There must be something I get wrong with. Could you correct my view?

2. How can we prove that $V$ contains some $\pi^{m}\mathcal O_L$?

Answer 1

It's better to take $\pi$ to be a prime element of $\mathcal{O}_K$ (probably a misprint in the notes).

For 2, note that for each $a\in L$, there exists an $m\geq 0$ such that $\pi^m a\in V$ --- to see this write $a=\sum c_{\tau}x_{\tau}$ with $c_{\tau}\in K$ and choose an $m$ such that $\pi^m c_{\tau}\in \mathcal{O}_K$ for all $\tau$. Now choose a basis for $\mathcal{O}_L$ as an $\mathcal{O}_K$-module, and choose an $m$ that works for every element of the basis.