Is the intersection of ramification groups in upper numbering of a $p$-adic local field trivial?

Question

Let $K$ be a $p$-adic local field, for example $\mathbb{Q}_p$. Let $G$ be the absolute Galois group of $K$, and let $G^v$($v\ge -1$) be the ramification groups in upper numbering, then is it true that $\bigcap_{v=0}^\infty G^v=\{0\}$?

Answer 1

Yes, at least if the upper ramification groups $G^\nu$ are defined as $\varprojlim_L\mathrm{Gal}(L/K)^\nu$ for $L/K$ finite Galois (e.g. as in [1]). This makes sense because the upper-numbering is compatible with quotients. Then it follows from the fact that for each finite $L/K$, $\mathrm{Gal}(L/K)^\nu$ is trivial for $\nu$ large enough ([2], IV.3).

[1]: Fontaine, Jean-Marc. Il n'y a pas de variété abélienne sur Z. Invent. 1985

[2]: Serre, Jean-Pierre. Corps Locaux