Ramification group#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.{{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 }}{{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 }}

The structure of the set of extensions is known better when L/K is Galois.

Let (K, v) be a valued field and let L be a finite Galois extension of K. Let S_v 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 S_v by σ[w] = [w ∘ σ] (i.e. w is a representative of the equivalence class [w] ∈ S_v and [w] is sent to the equivalence class of the 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.

Given a fixed extension w of v to L, the decomposition group of w is the stabilizer subgroup G_w of [w], i.e. it is the subgroup of G consisting of all elements that fix the equivalence class [w] ∈ S_v.

Let m_w denote the maximal ideal of w inside the valuation ring R_w of w. The inertia group of w is the subgroup I_w of G_w consisting of elements σ such that σx ≡ x (mod m_w) for all x in R_w. In other words, I_w consists of the elements of the decomposition group that act trivially on the residue field of w. It is a normal subgroup of G_w.

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 are a refinement of the Galois group $G$ of a finite $L/K$ Galois extension of local fields. We shall write $w,\; \backslash mathcal\; O\_L,\; \backslash mathfrak\; p$ for the valuation, the ring of integers and its maximal ideal for $L$. As a consequence of Hensel's lemma, one can write $\backslash mathcal\; O\_L\; =\; \backslash mathcal\; O\_K[\backslash alpha]$ for some $\backslash alpha\; \backslash in\; L$ where $\backslash mathcal\; O\_K$ is the ring of integers of $K$.Neukirch (1999) p.178 (This is stronger than the primitive element theorem.) Then, for each integer $i\; \backslash ge\; -1$, we define $G\_i$ to be the set of all $s\; \backslash in\; G$ that satisfies the following equivalent conditions.

• (i) $s$ operates trivially on $\backslash mathcal\; O\_L\; /\; \backslash mathfrak\; p^\{i+1\}.$
• (ii) $w(s(x)\; -\; x)\; \backslash ge\; i+1$ for all $x\; \backslash in\; \backslash mathcal\; O\_L$
• (iii) $w(s(\backslash alpha)\; -\; \backslash alpha)\; \backslash ge\; i+1.$

The group $G\_i$ is called $i$-th ramification group. They form a decreasing filtration,

:$G\_\{-1\}\; =\; G\; \backslash supset\; G\_0\; \backslash supset\; G\_1\; \backslash supset\; \backslash dots\; \backslash \{*\backslash \}.$

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,

• $G/G\_0\; =\; \backslash operatorname\{Gal\}(l/k),$ where $l,\; k$ are the (finite) residue fields of $L,\; K$.since $G/G\_0$ is canonically isomorphic to the decomposition group.
• $G\_0\; =\; 1\; \backslash Leftrightarrow\; L/K$ is unramified.
• $G\_1\; =\; 1\; \backslash Leftrightarrow\; L/K$ 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 $i\; \backslash ge\; 0$.

One also defines the function $i\_G(s)\; =\; w(s(\backslash alpha)\; -\; \backslash alpha),\; s\; \backslash in\; G$. (ii) in the above shows $i\_G$ is independent of choice of $\backslash alpha$ 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 $s,\; t\; \backslash in\; G$,

• $i\_G(s)\; \backslash ge\; i\; +\; 1\; \backslash Leftrightarrow\; s\; \backslash in\; G\_i.$
• $i\_G(t\; s\; t^\{-1\})\; =\; i\_G(s).$
• $i\_G(st)\; \backslash ge\; \backslash min\backslash \{\; i\_G(s),\; i\_G(t)\; \backslash \}.$

Fix a uniformizer $\backslash pi$ of $L$. Then $s\; \backslash mapsto\; s(\backslash pi)/\backslash pi$ induces the injection $G\_i/G\_\{i+1\}\; \backslash to\; U\_\{L,\; i\}/U\_\{L,\; i+1\},\; i\; \backslash ge\; 0$ where $U\_\{L,\; 0\}\; =\; \backslash mathcal\{O\}\_L^\backslash times,\; U\_\{L,\; i\}\; =\; 1\; +\; \backslash mathfrak\{p\}^i$. (The map actually does not depend on the choice of the uniformizer.Conrad) It follows from thisUse $U\_\{L,\; 0\}/U\_\{L,\; 1\}\; \backslash simeq\; l^\backslash times$ and $U\_\{L,\; i\}/U\_\{L,\; i+1\}\; \backslash approx\; l^+$

• $G\_0/G\_1$ 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 $\backslash mathfrak\{D\}\_\{L/K\}$ of the extension $L/K$ and that of subextensions:Serre (1979) 4.1 Prop.4, p.64

:$w(\backslash mathfrak\{D\}\_\{L/K\})\; =\; \backslash sum\_\{s\; \backslash ne\; 1\}\; i\_G(s)\; =\; \backslash sum\_\{i=0\}^\backslash infty\; (|G\_i|\; -\; 1).$

If $H$ is a normal subgroup of $G$, then, for $\backslash sigma\; \backslash in\; G$, $i\_\{G/H\}(\backslash sigma)\; =\; \{1\; \backslash over\; e\_\{L/K\}\}\; \backslash sum\_\{s\; \backslash mapsto\; \backslash sigma\}\; i\_G(s)$.Serre (1979) 4.1. Prop.3, p.63

Combining this with the above one obtains: for a subextension $F/K$ corresponding to $H$,

:$v\_F(\backslash mathfrak\{D\}\_\{F/K\})\; =\; \{1\; \backslash over\; e\_\{L/F\}\}\; \backslash sum\_\{s\; \backslash not\backslash in\; H\}\; i\_G(s).$

If $s\; \backslash in\; G\_i,\; t\; \backslash in\; G\_j,\; i,\; j\; \backslash ge\; 1$, then $sts^\{-1\}t^\{-1\}\; \backslash in\; G\_\{i+j+1\}$.Serre (1979) 4.2. Proposition 10. In the terminology of Lazard, this can be understood to mean the Lie algebra $\backslash operatorname\{gr\}(G\_1)\; =\; \backslash sum\_\{i\; \backslash ge\; 1\}\; G\_i/G\_\{i+1\}$ is abelian.

The ramification groups for a cyclotomic extension $K\_n\; :=\; \backslash mathbf\; Q\_p(\backslash zeta)/\backslash mathbf\; Q\_p$, where $\backslash zeta$ is a $p^n$-th primitive root of unity, can be described explicitly:Serre, Corps locaux. Ch. IV, §4, Proposition 18

:$G\_s\; =\; \backslash operatorname\{Gal\}(K\_n\; /\; K\_e),$

where e is chosen such that .

Let K be the extension of {{math|Q₂}} generated by $x\_1=\backslash sqrt\{2+\backslash sqrt\{2\}\}$. The conjugates of $x\_1$ are $x\_2\; =\; \backslash sqrt\{2-\backslash sqrt\{2\}\}$, $x\_3\; =\; -x\_1$, $x\_4\; =\; -x\_2$.

A little computation shows that the quotient of any two of these is a unit. Hence they all generate the same ideal; call it {{pi}}. $\backslash sqrt\{2\}$ generates {{pi}}²; (2)={{pi}}⁴.

Now $x\_1-x\_3=2x\_1$, which is in {{pi}}⁵.

and $x\_1\; -\; x\_2\; =\; \backslash sqrt\{4-2\backslash sqrt\{2\}\},$

which is in {{pi}}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 $G\_3\; =\; G\_4=(13)(24).$

$w(\backslash mathfrak\{D\}\_\{K/Q\_2\})\; =\; 3+3+3+1+1\; =\; 11,$ so that the different $\backslash mathfrak\{D\}\_\{K/Q\_2\}\; =\; \backslash pi^\{11\}$

$x\_1$ satisfies X⁴ − 4X² + 2, which has discriminant 2048 = 2¹¹.

If $u$ is a real number $\backslash ge\; -1$, let $G\_u$ denote $G\_i$ where i the least integer $\backslash ge\; u$. In other words, $s\; \backslash in\; G\_u\; \backslash Leftrightarrow\; i\_G(s)\; \backslash ge\; u+1.$ Define $\backslash phi$ bySerre (1967) p.156

:$\backslash phi(u)\; =\; \backslash int\_0^u\; \{dt\; \backslash over\; (G\_0\; :\; G\_t)\}$

where, by convention, $(G\_0\; :\; G\_t)$ is equal to $(G\_\{-1\}\; :\; G\_0)^\{-1\}$ if $t\; =\; -1$ and is equal to $1$ for .Neukirch (1999) p.179 Then $\backslash phi(u)\; =\; u$ for $-1\; \backslash le\; u\; \backslash le\; 0$. It is immediate that $\backslash phi$ is continuous and strictly increasing, and thus has the continuous inverse function $\backslash psi$ defined on $[-1,\; \backslash infty)$. Define

$G^v\; =\; G\_\{\backslash psi(v)\}$.

$G^v$ is then called the v-th ramification group in upper numbering. In other words, $G^\{\backslash phi(u)\}\; =\; G\_u$. Note $G^\{-1\}\; =\; G,\; G^0\; =\; G\_0$. The upper numbering is defined so as to be compatible with passage to quotients: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 states that the ramification groups in the lower numbering satisfy $G\_u\; H/H\; =\; (G/H)\_v$ (for $v\; =\; \backslash phi\_\{L/F\}(u)$ where $L/F$ is the subextension corresponding to $H$), and that the ramification groups in the upper numbering satisfy $G^u\; H/H\; =\; (G/H)^u$.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., $G\_i\; =\; G\_\{i+1\}$ whenever $\backslash phi(i)$ 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. The image of $G^n(L/K)$ under the isomorphism

:$G(L/K)^\{\backslash mathrm\{ab\}\}\; \backslash leftrightarrow\; K^*/N\_\{L/K\}(L^*)$

is justSnaith (1994) pp.30-31

:$U^n\_K\; /\; (U^n\_K\; \backslash cap\; N\_\{L/K\}(L^*))\; \backslash \; .$

• Finite extensions of local fields

{{reflist|2}}

• 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