Inmathematics, the ramification theory of valuations studies the set of extensions of a valuationv of a fieldK to an extensionLofK. It is a generalization of the ramification theory of Dedekind domains.[1][2]
The structure of the set of extensions is known better when L/KisGalois.
Let (K, v) be a valued field and let L be a finiteGalois extensionofK. Let Sv be the set of equivalenceclasses of extensions of vtoL and let G be the Galois groupofL over K. Then G acts on Sv by σ[w] = [w ∘ σ] (i.e. w is a representative of the equivalence class [w] ∈ Sv and [w] is sent to the equivalence class of the compositionofw with the automorphismσ : L → L; this is independent of the choice of w in [w]). In fact, this action is transitive.
Given a fixed extension wofvtoL, the decomposition group of w is the stabilizer subgroupGw of [w], i.e. it is the subgroupofG consisting of all elements that fix the equivalence class [w] ∈ Sv.
Let mw denote the maximal idealofw inside the valuation ringRwofw. The inertia group of w is the subgroup IwofGw consisting of elements σ such that σx ≡ x (mod mw) for all xinRw. In other words, Iw consists of the elements of the decomposition group that act trivially on the residue fieldofw. It is a normal subgroupofGw.
Ramification groups are a refinement of the Galois group of a finite Galois extensionoflocal fields. We shall write for the valuation, the ring of integers and its maximal ideal for . As a consequence of Hensel's lemma, one can write for some where is the ring of integers of .[3] (This is stronger than the primitive element theorem.) Then, for each integer , we define to be the set of all that satisfies the following equivalent conditions.
(i) operates trivially on
(ii) for all
(iii)
The group is called -th ramification group. They form a decreasing filtration,
In fact, the are normal by (i) and trivial for sufficiently large by (iii). For the lowest indices, it is customary to call the inertia subgroupof because of its relation to splitting of prime ideals, while the wild inertia subgroupof. The quotient is called the tame quotient.
The Galois group and its subgroups are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular,
istamely ramified (i.e., the ramification index is prime to the residue characteristic.)
The study of ramification groups reduces to the totally ramified case since one has for .
One also defines the function . (ii) in the above shows is independent of choice of and, moreover, the study of the filtration is essentially equivalent to that of .[5] satisfies the following: for ,
Fix a uniformizer of. Then induces the injection where . (The map actually does not depend on the choice of the uniformizer.[6]) It follows from this[7]
If is a real number , let denote where i the least integer . In other words, Define by[12]
where, by convention, is equal to if and is equal to for .[13] Then for . It is immediate that is continuous and strictly increasing, and thus has the continuous inverse function defined on . Define
.
is then called the v-th ramification group in upper numbering. In other words, . Note . The upper numbering is defined so as to be compatible with passage to quotients:[14]if is normal in , then
for all
(whereas lower numbering is compatible with passage to subgroups.)
Herbrand's theorem states that the ramification groups in the lower numbering satisfy (for where is the subextension corresponding to ), and that the ramification groups in the upper numbering satisfy .[15][16] This allows one to define ramification groups in the upper numbering for infinite Galois extensions (such as the absolute Galois group of a local field) from the inverse system of ramification groups for finite subextensions.
The upper numbering for an abelian extension is important because of the Hasse–Arf theorem. It states that if is abelian, then the jumps in the filtration are integers; i.e., whenever is not an integer.[17]
The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the Artin isomorphism. The image of under the isomorphism
Serre, Jean-Pierre (1967). "VI. Local class field theory". In Cassels, J.W.S.; Fröhlich, A. (eds.). Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. London: Academic Press. pp. 128–161. Zbl0153.07403.