Ramification group

In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains. The structure of the set of extensions is known better when L/K is Galois. Decomposition group and inertia group Let (K, v) be a valued field and let L be a finite Galois extension of K. Let Sv be the set of equivalence classes of extensions of v to L and let G be the Galois group of L 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 composition of w with the automorphism ; this is independent of the choice of w in [w]). In fact, this action is transitive. Given a fixed extension w of v to L, the 'decomposition group of w''' is the stabilizer subgroup Gw of [w], i.e. it is the subgroup of G consisting of all elements that fix the equivalence class [w] ∈ Sv''. Let mw denote the maximal ideal of w inside the valuation ring Rw of w. The 'inertia group of w''' is the subgroup Iw of Gw consisting of elements σ such that σx ≡ x (mod mw) for all x in Rw. In other words, Iw consists of the elements of the decomposition group that act trivially on the residue field of w. It is a normal subgroup of Gw''. The reduced ramification index e(w/v) is independent of w and is denoted e(v). Similarly, the relative degree f(w/v) is also independent of w and is denoted f(v). == Ramification groups in lower numbering ==

Ramification groups are a refinement of the Galois group G of a finite L/K Galois extension of local fields. We shall write w, \mathcal O_L, \mathfrak p for the valuation, the ring of integers and its maximal ideal for L. As a consequence of Hensel's lemma, one can write \mathcal O_L = \mathcal O_K[\alpha] for some \alpha \in L where \mathcal O_K is the ring of integers of K. (This is stronger than the primitive element theorem.) Then, for each integer i \ge -1, we define G_i to be the set of all s \in G that satisfies the following equivalent conditions. • (i) s operates trivially on \mathcal O_L / \mathfrak p^{i+1}. • (ii) w(s(x) - x) \ge i+1 for all x \in \mathcal O_L • (iii) w(s(\alpha) - \alpha) \ge i+1. The group G_i is called i-th ramification group. They form a decreasing filtration, :G_{-1} = G \supset G_0 \supset G_1 \supset \dots \{*\}. In fact, the G_i are normal by (i) and trivial for sufficiently large i by (iii). For the lowest indices, it is customary to call G_0 the inertia subgroup of G because of its relation to splitting of prime ideals, while G_1 the wild inertia subgroup of G. The quotient G_0 / G_1 is called the tame quotient. The Galois group G and its subgroups G_i are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular, • G/G_0 = \operatorname{Gal}(l/k), where l, k are the (finite) residue fields of L, K. • G_0 = 1 \Leftrightarrow L/K is unramified. • G_1 = 1 \Leftrightarrow L/K is tamely 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 G_i = (G_0)_i for i \ge 0. One also defines the function i_G(s) = w(s(\alpha) - \alpha), s \in G. (ii) in the above shows i_G is independent of choice of \alpha and, moreover, the study of the filtration G_i is essentially equivalent to that of i_G. i_G satisfies the following: for s, t \in G, • i_G(s) \ge i + 1 \Leftrightarrow s \in G_i. • i_G(t s t^{-1}) = i_G(s). • i_G(st) \ge \min\{ i_G(s), i_G(t) \}. Fix a uniformizer \pi of L. Then s \mapsto s(\pi)/\pi induces the injection G_i/G_{i+1} \to U_{L, i}/U_{L, i+1}, i \ge 0 where U_{L, 0} = \mathcal{O}_L^\times, U_{L, i} = 1 + \mathfrak{p}^i. (The map actually does not depend on the choice of the uniformizer.) It follows from this • G_0/G_1 is cyclic of order prime to p • G_i/G_{i+1} is a product of cyclic groups of order p. In particular, G_1 is a p-group and G_0 is solvable. The ramification groups can be used to compute the different \mathfrak{D}_{L/K} of the extension L/K and that of subextensions: :w(\mathfrak{D}_{L/K}) = \sum_{s \ne 1} i_G(s) = \sum_{i=0}^\infty (|G_i| - 1). If H is a normal subgroup of G, then, for \sigma \in G, i_{G/H}(\sigma) = {1 \over e_{L/K}} \sum_{s \mapsto \sigma} i_G(s). Combining this with the above one obtains: for a subextension F/K corresponding to H, :v_F(\mathfrak{D}_{F/K}) = {1 \over e_{L/F}} \sum_{s \not\in H} i_G(s). If s \in G_i, t \in G_j, i, j \ge 1, then sts^{-1}t^{-1} \in G_{i+j+1}. In the terminology of Lazard, this can be understood to mean the Lie algebra \operatorname{gr}(G_1) = \sum_{i \ge 1} G_i/G_{i+1} is abelian. Example: the cyclotomic extension The ramification groups for a cyclotomic extension K_n := \mathbf Q_p(\zeta)/\mathbf Q_p, where \zeta is a p^n-th primitive root of unity, can be described explicitly: :G_s = \operatorname{Gal}(K_n / K_e), where e is chosen such that p^{e-1} \le s . Example: a quartic extension Let K be the extension of generated by x_1=\sqrt{2+\sqrt{2}}. The conjugates of x_1 are x_2 = \sqrt{2-\sqrt{2}}, x_3 = -x_1, x_4 = -x_2. A little computation shows that the quotient of any two of these is a unit. Hence they all generate the same ideal; call it . \sqrt{2} generates 2; (2)=4. Now x_1-x_3=2x_1, which is in 5. and x_1 - x_2 = \sqrt{4-2\sqrt{2}}, which is in 3. Various methods show that the Galois group of K is C_4, cyclic of order 4. Also: : G_0 = G_1 = G_2 = C_4. and G_3 = G_4=(13)(24). w(\mathfrak{D}_{K/Q_2}) = 3+3+3+1+1 = 11, so that the different \mathfrak{D}_{K/Q_2} = \pi^{11} x_1 satisfies X4 − 4X2 + 2, which has discriminant 2048 = 211. == Ramification groups in upper numbering ==

If u is a real number \ge -1, let G_u denote G_i where i the least integer \ge u. In other words, s \in G_u \Leftrightarrow i_G(s) \ge u+1. Define \phi by :\phi(u) = \int_0^u {dt \over (G_0 : G_t)} where, by convention, (G_0 : G_t) is equal to (G_{-1} : G_0)^{-1} if t = -1 and is equal to 1 for -1 . Then \phi(u) = u for -1 \le u \le 0. It is immediate that \phi is continuous and strictly increasing, and thus has the continuous inverse function \psi defined on [-1, \infty). Define G^v = G_{\psi(v)}. G^v is then called the '''v-th ramification group''' in upper numbering. In other words, G^{\phi(u)} = G_u. Note G^{-1} = G, G^0 = G_0. The upper numbering is defined so as to be compatible with passage to quotients: if H is normal in G, then :(G/H)^v = G^v H / H for all v (whereas lower numbering is compatible with passage to subgroups.) Herbrand's theorem '''Herbrand's theorem''' states that the ramification groups in the lower numbering satisfy G_u H/H = (G/H)_v (for v = \phi_{L/F}(u) where L/F is the subextension corresponding to H), and that the ramification groups in the upper numbering satisfy G^u H/H = (G/H)^u. 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 G is abelian, then the jumps in the filtration G^v are integers; i.e., G_i = G_{i+1} whenever \phi(i) is not an integer. The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the Artin isomorphism. The image of G^n(L/K) under the isomorphism : G(L/K)^{\mathrm{ab}} \leftrightarrow K^*/N_{L/K}(L^*) is just : U^n_K / (U^n_K \cap N_{L/K}(L^*)) \ . ==See also==