Inertia Group Of An Extension Of Valuations
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In number theory, more specifically in local class field theory, the ramification groups are a
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension.


Ramification theory of valuations

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the ramification theory of valuations studies the set of extensions of a valuation ''v'' of a field ''K'' to an
extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
''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 Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *'' Equival ...
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''nbsp;=  'w'' ∘ σ(i.e. ''w'' is a representative of the equivalence class 'w''nbsp;∈ ''Sv'' and 'w''is sent to the equivalence class of the composition of ''w'' with the
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
; 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 In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
''Gw'' of 'w'' i.e. it is the subgroup of ''G'' consisting of all elements that fix the equivalence class 'w''nbsp;∈ ''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 σ''x'' ≡ ''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 In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
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 Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , whic ...
/math> for some \alpha \in L where \mathcal O_K is the ring of integers of K.Neukirch (1999) p.178 (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^. *(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 Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
, :G_ = 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 In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension. Ramificati ...
of G because of its relation to splitting of prime ideals, while G_1 the
wild inertia subgroup Wild, wild, wilds or wild may refer to: Common meanings * Wildlife, Wild animal * Wilderness, a wild natural environment * Wildness, the quality of being wild or untamed Art, media and entertainment Film and television * Wild (2014 film), ''Wi ...
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(l/k), where l, k are the (finite) residue fields of L, K. *G_0 = 1 \Leftrightarrow L/K is
unramified In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
. *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.Serre (1979) p.62 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^) = i_G(s). *i_G(st) \ge \min\. Fix a uniformizer \pi of L. Then s \mapsto s(\pi)/\pi induces the injection G_i/G_ \to U_/U_, i \ge 0 where U_ = \mathcal_L^\times, U_ = 1 + \mathfrak^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_ 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_ of the extension L/K and that of subextensions:Serre (1979) 4.1 Prop.4, p.64 :w(\mathfrak_) = \sum_ i_G(s) = \sum_^\infty (, G_i, - 1). If H is a normal subgroup of G, then, for \sigma \in G, i_(\sigma) = \sum_ i_G(s).Serre (1979) 4.1. Prop.3, p.63 Combining this with the above one obtains: for a subextension F/K corresponding to H, :v_F(\mathfrak_) = \sum_ i_G(s). If s \in G_i, t \in G_j, i, j \ge 1, then sts^t^ \in G_. In the terminology of Lazard, this can be understood to mean the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
\operatorname(G_1) = \sum_ G_i/G_ is abelian.


Example: the cyclotomic extension

The ramification groups for a
cyclotomic extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois ...
K_n := \mathbf Q_p(\zeta)/\mathbf Q_p, where \zeta is a p^n-th primitive root of unity, can be described explicitly:Serre, ''Corps locaux''. Ch. IV, §4, Proposition 18 :G_s = Gal(K_n / K_e), where ''e'' is chosen such that p^ \le s < p^e.


Example: a quartic extension

Let K be the extension of generated by x_1=\sqrt. The conjugates of x1 are x2=x_2 = \sqrt, ''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 generates 2; (2)=4. Now ''x''1 − ''x''3 = 2''x''1, which is in 5. and x_1 - x_2 = \sqrt, 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_) = 3+3+3+1+1 = 11, so that the different \mathfrak_ = \pi^ ''x''1 satisfies ''x''4 − 4''x''2 + 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 bySerre (1967) p.156 :\phi(u) = \int_0^u where, by convention, (G_0 : G_t) is equal to (G_ : G_0)^ if t = -1 and is equal to 1 for -1 < t \le 0.Neukirch (1999) p.179 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_. G^v is then called the ''v''-th ramification group in upper numbering. In other words, G^ = G_u. Note G^ = G, G^0 = G_0. The upper numbering is defined so as to be compatible with passage to quotients:Serre (1967) p.155 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_(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.Neukirch (1999) p.180Serre (1979) p.75 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_ whenever \phi(i) is not an integer.Neukirch (1999) p.355 The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the
Artin isomorphism Artin may refer to: * Artin (name), a surname and given name, including a list of people with the name ** Artin, a variant of Harutyun Harutyun ( hy, Հարություն and in Western Armenian Յարութիւն) also spelled Haroutioun, Harut ...
. The image of G^n(L/K) under the isomorphism : G(L/K)^ \leftrightarrow K^*/N_(L^*) is justSnaith (1994) pp.30-31 : U^n_K / (U^n_K \cap N_(L^*)) \ .


See also

* Finite extensions of local fields


Notes


References

*B. Conrad
Math 248A. Higher ramification groups
* * * * * {{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 Algebraic number theory