Ramification group of valuations - need terminology

[coquelicot]

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 α > 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 :

* { σ ∈ Gal(L/K) : w(σ x - x) ≥ α },

* {σ ∈ Gal(L/K) : w(σ x - x) > α },

* { σ ∈ Gal(L/K) : w(σ x - x) ≥ w(x) + α },

* { σ ∈ Gal(L/K) : w(σ x - x) > w(x) + α}.

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

 

[jvicens]

Dear forum poster,

Thank you for your question. The terminology for these subgroups is indeed "ramification group of order α". These subgroups are also known as the higher ramification groups, as they correspond to higher levels of ramification in the Galois extension L/K.

As for sources, I would recommend looking at the book "Local Fields" by Jean-Pierre Serre, which discusses the theory of valuations and ramification groups in depth. Additionally, the book "Algebraic Number Theory" by Jürgen Neukirch also covers this topic extensively. Both of these sources should provide you with a thorough understanding of the terminology and concepts related to ramification groups.

I hope this helps. Happy reading!