I've asked this question in several more elementary forums, and haven't get any answer. So I presume this is not so an elementary question.

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 sources where these groups are dealt in ? Thx.