Filtration of the Galois group of a local field extension
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.
Ramification theory of valuations
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.[1][2]
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 σ : L → L; 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
of a finite
Galois extension of local 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 ![{\displaystyle {\mathcal {O}}_{L}/{\mathfrak {p}}^{i+1}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- (ii)
for all ![{\displaystyle x\in {\mathcal {O}}_{L}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- (iii)
![{\displaystyle w(s(\alpha )-\alpha )\geq i+1.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The group
is called
-th ramification group. They form a decreasing filtration,
![{\displaystyle G_{-1}=G\supset G_{0}\supset G_{1}\supset \dots \{*\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 subgroup of
because of its relation to splitting of prime ideals, while
the wild inertia subgroup of
. 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,
where
are the (finite) residue fields of
.[4]
is unramified.
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
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
,
![{\displaystyle i_{G}(s)\geq i+1\Leftrightarrow s\in G_{i}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle i_{G}(tst^{-1})=i_{G}(s).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle i_{G}(st)\geq \min\{i_{G}(s),i_{G}(t)\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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]
is cyclic of order prime to ![{\displaystyle p}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is a product of cyclic groups of order
.
In particular,
is a p-group and
is solvable.
The ramification groups can be used to compute the different
of the extension
and that of subextensions:[8]
![{\displaystyle w({\mathfrak {D}}_{L/K})=\sum _{s\neq 1}i_{G}(s)=\sum _{i=0}^{\infty }(|G_{i}|-1).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
If
is a normal subgroup of
, then, for
,
.[9]
Combining this with the above one obtains: for a subextension
corresponding to
,
![{\displaystyle v_{F}({\mathfrak {D}}_{F/K})={1 \over e_{L/F}}\sum _{s\not \in H}i_{G}(s).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
If
, then
.[10] In the terminology of Lazard, this can be understood to mean the Lie algebra
is abelian.
Example: the cyclotomic extension
The ramification groups for a cyclotomic extension
, where
is a
-th primitive root of unity, can be described explicitly:[11]
![{\displaystyle G_{s}=\operatorname {Gal} (K_{n}/K_{e}),}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where e is chosen such that
.
Example: a quartic extension
Let K be the extension of Q2 generated by
. The conjugates of
are
,
,
.
A little computation shows that the quotient of any two of these is a unit. Hence they all generate the same ideal; call it π.
generates π2; (2)=π4.
Now
, which is in π5.
and
which is in π3.
Various methods show that the Galois group of K is
, cyclic of order 4. Also:
![{\displaystyle G_{0}=G_{1}=G_{2}=C_{4}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and ![{\displaystyle G_{3}=G_{4}=(13)(24).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
so that the different ![{\displaystyle {\mathfrak {D}}_{K/Q_{2}}=\pi ^{11}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
satisfies X4 − 4X2 + 2, which has discriminant 2048 = 211.
Ramification groups in upper numbering
If
is a real number
, let
denote
where i the least integer
. In other words,
Define
by[12]
![{\displaystyle \phi (u)=\int _{0}^{u}{dt \over (G_{0}:G_{t})}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 ![{\displaystyle v}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
(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
(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
![{\displaystyle G(L/K)^{\mathrm {ab} }\leftrightarrow K^{*}/N_{L/K}(L^{*})}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is just[18]
![{\displaystyle U_{K}^{n}/(U_{K}^{n}\cap N_{L/K}(L^{*}))\ .}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
See also
Notes
- ^ Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. Vol. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001.
- ^ Zariski, Oscar; Samuel, Pierre (1976) [1960]. Commutative algebra, Volume II. Graduate Texts in Mathematics. Vol. 29. New York, Heidelberg: Springer-Verlag. Chapter VI. ISBN 978-0-387-90171-8. Zbl 0322.13001.
- ^ Neukirch (1999) p.178
- ^ since
is canonically isomorphic to the decomposition group. - ^ Serre (1979) p.62
- ^ Conrad
- ^ Use
and ![{\displaystyle U_{L,i}/U_{L,i+1}\approx l^{+}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- ^ Serre (1979) 4.1 Prop.4, p.64
- ^ Serre (1979) 4.1. Prop.3, p.63
- ^ Serre (1979) 4.2. Proposition 10.
- ^ Serre, Corps locaux. Ch. IV, §4, Proposition 18
- ^ Serre (1967) p.156
- ^ Neukirch (1999) p.179
- ^ Serre (1967) p.155
- ^ Neukirch (1999) p.180
- ^ Serre (1979) p.75
- ^ Neukirch (1999) p.355
- ^ Snaith (1994) pp.30-31
References
- B. Conrad, Math 248A. Higher ramification groups
- Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. Vol. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001.
- Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
- 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. Zbl 0153.07403.
- Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. Vol. 67. Translated by Greenberg, Marvin Jay. Berlin, New York: Springer-Verlag. ISBN 0-387-90424-7. MR 0554237. Zbl 0423.12016.
- Snaith, Victor P. (1994). Galois module structure. Fields Institute monographs. Providence, RI: American Mathematical Society. ISBN 0-8218-0264-X. Zbl 0830.11042.