I am lost and need some terminology (also hopefully sources). Let $L/K$ be a Galois extension, and $w$ be a valuation of a $L$, lying above a valuation $v$ of $K$. Notice that I do not suppose that $w$ is discrete. Given $\alpha >0$ in the finite image of $w$, each of the following can easily been shown to be a subgroup of the inertia group of $w$ in $L$ :

  • $\{ \sigma\in Gal(L/K) : w(\sigma x - x) \geq \alpha \}$,

  • $\{ \sigma\in Gal(L/K) : w(\sigma x - x) > \alpha \}$,

  • $\{ \sigma\in Gal(L/K) : w(\sigma x - x) \geq w(x) + \alpha \}$,

  • $\{ \sigma\in Gal(L/K) : w(\sigma x - x) > w(x) + \alpha \}$.

What is the terminology for these subgroups ? (I guess some variant of "ramification group of order $\alpha$) ? Can you indicate me a source ? Thx.

  • $\begingroup$ When the valuations are discrete and $L/K$ is an extension of local fields, the first two groups you indicated are called "higher ramification groups". $\endgroup$ – Ferra Dec 18 '14 at 11:14
  • $\begingroup$ Unfortunately, this does not help me because I do not suppose the valuations are discrete, which is all the point (since the two first groups are allegedly not of the same kind in general). $\endgroup$ – MikeTeX Dec 18 '14 at 11:39

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