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 $\bigcap_{v=0}^\infty G^v=\{0\}$?


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

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  • $\begingroup$ Nice answer !!! $\endgroup$ – Bonbon May 13 '18 at 18:24

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