Editing Ramification group

{{Short description|Filtration of the Galois group of a local field extension}}
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In [[number theory]], more specifically in [[local class field theory]], the '''ramification groups''' are a [[Filtration (mathematics)|filtration]] of the [[Galois group]] of a [[local field]] extension, which gives detailed information on the [[Ramification (mathematics)|ramification]] phenomena of the extension.

==Ramification theory of valuations==
In [[mathematics]], the '''ramification theory of valuations''' studies the set of [[extension of a valuation|extensions]] of a [[valuation (algebra)|valuation]] ''v'' of a [[Field (mathematics)|field]] ''K'' to an [[field extension|extension]] ''L'' of ''K''. It is a generalization of the ramification theory of Dedekind domains.<ref>{{cite book | last1=Fröhlich | first1=A. | author1-link=Albrecht Fröhlich | last2=Taylor | first2= M.J. | author2-link=Martin J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | publisher=[[Cambridge University Press]] | year=1991 | isbn=0-521-36664-X | zbl=0744.11001 }}</ref><ref>{{cite book | last=Zariski | first=Oscar | author-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative algebra, Volume II | publisher=Springer-Verlag | location=New York, Heidelberg | series=[[Graduate Texts in Mathematics]] | volume=29 | year=1976 | origyear=1960 | isbn=978-0-387-90171-8 | zbl=0322.13001 | at=Chapter VI }}</ref>

The structure of the set of extensions is known better when ''L''/''K'' is [[Galois extension|Galois]].

===<span id="decomp"></span><span id="inertia"></span>Decomposition group and inertia group===

Let (''K'',&nbsp;''v'') be a [[valued field]] and let ''L'' be a [[finite extension|finite]] [[Galois extension]] of ''K''. Let ''S<sub>v</sub>'' be the set of [[equivalence of valuations|equivalence]] [[equivalence class|classes]] of extensions of ''v'' to ''L'' and let ''G'' be the [[Galois group]] of ''L'' over ''K''. Then ''G'' acts on ''S<sub>v</sub>'' by σ[''w'']&nbsp;=&nbsp;[''w''&nbsp;∘&nbsp;σ] (i.e. ''w'' is a [[representative (mathematics)|representative]] of the equivalence class [''w'']&nbsp;∈&nbsp;''S<sub>v</sub>'' and [''w''] is sent to the equivalence class of the [[function composition|composition]] of ''w'' with the [[automorphism]] {{nowrap|σ : ''L'' → ''L''}}; this is independent of the choice of ''w'' in [''w'']). In fact, this action is [[transitive action|transitive]].

Given a fixed extension ''w'' of ''v'' to ''L'', the '''decomposition group of ''w''''' is the [[stabilizer subgroup]] ''G<sub>w</sub>'' of [''w''], i.e. it is the [[subgroup]] of ''G'' consisting of all elements that fix the equivalence class [''w'']&nbsp;∈&nbsp;''S<sub>v</sub>''.

Let ''m<sub>w</sub>'' denote the [[maximal ideal of a valuation|maximal ideal]] of ''w'' inside the [[valuation ring of a valuation|valuation ring]] ''R<sub>w</sub>'' of ''w''. The '''inertia group of ''w''''' is the subgroup ''I<sub>w</sub>'' of ''G<sub>w</sub>'' consisting of elements ''σ'' such that σ''x''&nbsp;≡&nbsp;''x''&nbsp;(mod&nbsp;''m<sub>w</sub>'') for all ''x'' in ''R<sub>w</sub>''. In other words, ''I<sub>w</sub>'' consists of the elements of the decomposition group that [[trivial action|act trivially]] on the [[residue field of a valuation|residue field]] of ''w''. It is a [[normal subgroup]] of ''G<sub>w</sub>''.

The [[Reduced ramification index of an extension of valuations|reduced ramification index]] ''e''(''w''/''v'') is independent of ''w'' and is denoted ''e''(''v''). Similarly, the [[Relative degree of an extension of valuations|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  <math>G</math> of a finite <math>L/K</math> [[Galois extension]] of [[local field]]s. We shall write <math>w, \mathcal O_L, \mathfrak p</math> for the valuation, the ring of integers and its maximal ideal for <math>L</math>. As a consequence of [[Hensel's lemma]], one can write <math>\mathcal O_L = \mathcal O_K[\alpha]</math> for some <math>\alpha \in L</math> where <math>\mathcal O_K</math> is the ring of integers of <math>K</math>.<ref name=N178>Neukirch (1999) p.178</ref> (This is stronger than the [[primitive element theorem]].) Then, for each integer <math>i \ge -1</math>, we define <math>G_i</math> to be the set of all <math>s \in G</math> that satisfies the following equivalent conditions.
*(i) <math>s</math> operates trivially on <math>\mathcal O_L / \mathfrak p^{i+1}.</math>
*(ii) <math>w(s(x) - x) \ge i+1</math> for all <math>x \in \mathcal O_L</math>
*(iii) <math>w(s(\alpha) - \alpha) \ge i+1.</math>

The group <math>G_i</math> is called ''<math>i</math>-th ramification group''. They form a decreasing [[filtration (mathematics)|filtration]],
:<math>G_{-1} = G \supset G_0 \supset G_1 \supset \dots \{*\}.</math>
In fact, the  <math>G_i</math> are normal by (i) and [[trivial group|trivial]] for sufficiently large <math>i</math> by (iii). For the lowest indices, it is customary to call <math>G_0</math> the [[inertia subgroup]] of <math>G</math> because of its relation to [[Splitting of prime ideals in Galois extensions|splitting of prime ideals]], while <math>G_1</math> the [[wild inertia subgroup]] of <math>G</math>. The quotient <math>G_0 / G_1</math> is called the tame quotient.

The Galois group <math>G</math> and its subgroups <math>G_i</math> are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular,
*<math>G/G_0 = \operatorname{Gal}(l/k),</math> where <math>l, k</math> are the (finite) residue fields of  <math>L, K</math>.<ref>since <math>G/G_0</math> is canonically isomorphic to the decomposition group.</ref>
*<math>G_0 = 1 \Leftrightarrow L/K </math> is [[unramified extension|unramified]].
*<math>G_1 = 1 \Leftrightarrow L/K </math> 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 <math>G_i = (G_0)_i</math> for <math>i \ge 0</math>.

One also defines the function <math>i_G(s) = w(s(\alpha) - \alpha), s \in G</math>. (ii) in the above shows <math>i_G</math> is independent of choice of <math>\alpha</math> and, moreover, the study of the filtration <math>G_i</math> is essentially equivalent to that of <math>i_G</math>.<ref name=S7962>Serre (1979) p.62</ref> <math>i_G</math> satisfies the following: for <math>s, t \in G</math>,
*<math>i_G(s) \ge i + 1 \Leftrightarrow s \in G_i.</math>
*<math>i_G(t s t^{-1}) = i_G(s).</math>
*<math>i_G(st) \ge \min\{ i_G(s), i_G(t) \}.</math>

Fix a uniformizer <math>\pi</math> of <math>L</math>. Then <math>s \mapsto s(\pi)/\pi</math> induces the injection <math>G_i/G_{i+1} \to U_{L, i}/U_{L, i+1}, i \ge 0</math> where <math>U_{L, 0} = \mathcal{O}_L^\times, U_{L, i} = 1 + \mathfrak{p}^i</math>. (The map actually does not depend on the choice of the uniformizer.<ref>Conrad</ref>) It follows from this<ref>Use <math>U_{L, 0}/U_{L, 1} \simeq l^\times</math> and <math>U_{L, i}/U_{L, i+1} \approx l^+</math></ref>
*<math>G_0/G_1</math> is cyclic of order prime to <math>p</math>
*<math>G_i/G_{i+1}</math> is a product of cyclic groups of order <math>p</math>.
In particular, <math>G_1</math> is a [[p-group|''p''-group]] and <math>G_0</math> is [[solvable group|solvable]].

The ramification groups can be used to compute the [[Different ideal|different]] <math>\mathfrak{D}_{L/K}</math> of the extension <math>L/K</math> and that of subextensions:<ref name=S64>Serre (1979) 4.1 Prop.4, p.64</ref>

:<math>w(\mathfrak{D}_{L/K}) = \sum_{s \ne 1} i_G(s) = \sum_{i=0}^\infty (|G_i| - 1).</math>

If <math>H</math> is a normal subgroup of <math>G</math>, then, for <math>\sigma \in G</math>, <math>i_{G/H}(\sigma) = {1 \over e_{L/K}} \sum_{s \mapsto \sigma} i_G(s)</math>.<ref name=S63>Serre (1979) 4.1. Prop.3, p.63</ref>

Combining this with the above one obtains: for a subextension <math>F/K</math> corresponding to <math>H</math>,
:<math>v_F(\mathfrak{D}_{F/K}) = {1 \over e_{L/F}} \sum_{s \not\in H} i_G(s).</math>

If <math>s \in G_i, t \in G_j, i, j \ge 1</math>, then <math>sts^{-1}t^{-1} \in G_{i+j+1}</math>.<ref>Serre (1979) 4.2. Proposition 10.</ref> In the terminology of [[Michel Lazard|Lazard]], this can be understood to mean the [[Lie algebra]] <math>\operatorname{gr}(G_1) = \sum_{i \ge 1} G_i/G_{i+1}</math> is abelian.

===Example: the cyclotomic extension===
The ramification groups for a [[cyclotomic extension]] <math>K_n := \mathbf Q_p(\zeta)/\mathbf Q_p</math>, where <math>\zeta</math> is a <math>p^n</math>-th primitive [[root of unity]], can be described explicitly:<ref>Serre, ''Corps locaux''. Ch. IV, §4, Proposition 18</ref>
:<math>G_s = \operatorname{Gal}(K_n / K_e),</math>
where ''e'' is chosen such that <math>p^{e-1} \le s < p^e</math>.

===Example: a quartic extension===
Let ''K'' be the extension of {{math|'''Q'''<sub>2</sub>}} generated by <math>x_1=\sqrt{2+\sqrt{2}}</math>. The conjugates of <math>x_1</math> are <math> x_2 = \sqrt{2-\sqrt{2}}</math>, <math>x_3 = -x_1</math>, <math>x_4 = -x_2</math>.

A little computation shows that the quotient of any two of these is a [[unit (ring theory)|unit]]. Hence they all generate the same ideal; call it {{pi}}. <math>\sqrt{2}</math> generates {{pi}}<sup>2</sup>; (2)={{pi}}<sup>4</sup>.

Now <math>x_1-x_3=2x_1</math>, which is in {{pi}}<sup>5</sup>.

and <math> x_1 - x_2 = \sqrt{4-2\sqrt{2}},
</math> which is in {{pi}}<sup>3</sup>.

Various methods show that the Galois group of ''K'' is <math>C_4</math>, cyclic of order 4. Also:

: <math>G_0 = G_1 = G_2 = C_4.</math>

and <math>G_3 = G_4=(13)(24). </math>

<math>w(\mathfrak{D}_{K/Q_2}) = 3+3+3+1+1 = 11,</math> so that the different <math>\mathfrak{D}_{K/Q_2} = \pi^{11} </math>

<math>x_1</math> satisfies ''X''<sup>4</sup> − 4''X''<sup>2</sup> + 2, which has discriminant 2048 = 2<sup>11</sup>.

== Ramification groups in upper numbering ==
If <math>u</math> is a real number <math>\ge -1</math>, let <math>G_u</math> denote <math>G_i</math> where ''i'' the least integer <math>\ge u</math>. In other words, <math>s \in G_u \Leftrightarrow i_G(s) \ge u+1.</math> Define <math>\phi</math> by<ref name=S67156>Serre (1967) p.156</ref>
:<math>\phi(u) = \int_0^u {dt \over (G_0 : G_t)}</math>
where, by convention, <math>(G_0 : G_t)</math> is equal to <math>(G_{-1} : G_0)^{-1}</math> if <math>t = -1</math> and is equal to <math>1</math> for <math>-1 < t \le 0</math>.<ref name=N179>Neukirch (1999) p.179</ref>  Then <math>\phi(u) = u</math> for <math>-1 \le u \le 0</math>. It is immediate that <math>\phi</math> is continuous and strictly increasing, and thus has the continuous inverse function <math>\psi</math> defined on <math>[-1, \infty)</math>. Define
<math>G^v = G_{\psi(v)}</math>.
<math>G^v</math> is then called the '''''v''-th ramification group''' in upper numbering. In other words, <math>G^{\phi(u)} = G_u</math>. Note <math>G^{-1} = G, G^0 = G_0</math>. The upper numbering is defined so as to be compatible with passage to quotients:<ref name=S67155>Serre (1967) p.155</ref> if <math>H</math> is normal in <math>G</math>, then
:<math>(G/H)^v = G^v H / H</math> for all <math>v</math>
(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 <math>G_u H/H = (G/H)_v</math> (for <math>v = \phi_{L/F}(u)</math> where <math>L/F</math> is the subextension corresponding to <math>H</math>), and that the ramification groups in the upper numbering satisfy <math>G^u H/H = (G/H)^u</math>.<ref name=N180>Neukirch (1999) p.180</ref><ref name=S75>Serre (1979) p.75</ref>  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 <math>G</math> is abelian, then the jumps in the filtration <math>G^v</math> are integers; i.e., <math>G_i = G_{i+1}</math> whenever <math>\phi(i)</math> is not an integer.<ref name=N355>Neukirch (1999) p.355</ref>

The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the [[Artin isomorphism]].  The image of <math>G^n(L/K)</math> under the isomorphism

:<math> G(L/K)^{\mathrm{ab}} \leftrightarrow K^*/N_{L/K}(L^*) </math>

is just<ref name=Sn3031>Snaith (1994) pp.30-31</ref>

:<math> U^n_K / (U^n_K \cap N_{L/K}(L^*)) \ . </math>

==See also==
*[[Finite extensions of local fields]]

== Notes ==
{{reflist|2}}

==References==
*B. Conrad, [http://math.stanford.edu/~conrad/248APage/handouts/ramgroup.pdf Math 248A. Higher ramification groups]
* {{cite book | last1=Fröhlich | first1=A. | author1-link=Albrecht Fröhlich | last2=Taylor | first2= M.J. | author2-link=Martin J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | publisher=[[Cambridge University Press]] | year=1991 | isbn=0-521-36664-X | zbl=0744.11001 }}
*{{Neukirch ANT}}
* {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | chapter=VI. Local class field theory | pages=128–161 | editor1-last=Cassels | editor1-first=J.W.S. | editor1-link=J. W. S. Cassels | editor2-last=Fröhlich | editor2-first=A. | editor2-link=Albrecht Fröhlich | title=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 | location=London | publisher=Academic Press | year=1967 | zbl=0153.07403 }} 
* {{cite book | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=[[Local Fields]]  | publisher=[[Springer-Verlag]] | location=Berlin, New York | mr=0554237 | year=1979 | translator-link1=Marvin Greenberg|translator-first1=Marvin Jay |translator-last1=Greenberg | series=Graduate Texts in Mathematics | volume=67 | isbn=0-387-90424-7 | zbl=0423.12016 }}
* {{cite book | last=Snaith | first=Victor P. | title=Galois module structure | series=Fields Institute monographs | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0264-X | zbl=0830.11042 }}

[[Category:Algebraic number theory]]