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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 20. Выпуск 2.
УДК 512.62 DOI 10.22405/2226-8383-2019-20-2-186-197
О двух подходах к классификации высших локальных полей
О. Ю. Иванова, С. В. Востоков, И. Б. Жуков
Иванова Ольга Юрьевна — Санкт-Петербургский государственный университет авиационного приборостроения, г. Санкт-Петербург. e-mail: olgaiv80@mail.ru
Востоков Сергей Владимирович — Санкт-Петербургский государственный университет, г. Санкт-Петербург. e-mail: s.vostokov@spbu.ru
Жуков Игорь Борисович — Санкт-Петербургский государственный университет, г. Санкт-Петербург.
e-mail: i.zhukov@spbu.ru
Аннотация
Эта статья связывает классификацию Курихары о полных дискретных оценочных полях и теории устранения дикого ветвления Эппа.
Для любого полного дискретного поля оценки К с произвольным полем вычетов простой характеристики можно определить некоторый численный инвариант Г(^), который лежит в основе классификации Курихары таких полей на 2 типа: поле К имеет тип I тогда и только тогда, когда Г (К) положительно. Значение этого инварианта указывает, насколько далеко данное поле от стандартного, т. е. от поля, которое неразветвлено над его постоянным подпол ем к, которое является максимальным подполем с совершенным полем вычетов. (Стандартные 2-мерные локальные поля являются точными полями вида
ММ}.)
Мы доказываем (при некотором мягком ограничении на К), что для смешанного характеристического 2-мерного локального поля типа I К существует оценка снизу для [/ : к], где 1/к является расширением, таким что 1К является стандартным полем (существующим из-за теории Ерр); логарифм этой степени может быть оценен линейно в терминах Г (К) с коэффициентом, зависящим только от ек/к.
Ключевые слова: Высшие локальные поля, дикое ветвление
Библиография: 15 названий.
Для цитирования:
О. Ю. Иванова, С. В. Востоков, И. Б. Жуков. О двух подходах к классификации высших локальных полей // Чебышевский сборник, 2019, т. 20, вып. 2, с. 186-197.
CHEBYSHEVSKII SBORNIK Vol. 20. No. 2.
UDC 512.62 DOI 10.22405/2226-8383-2019-20-2-186-197
On two approaches to classification of higher local fields1
O. Ivanova, S. Vostokov, I. Zhukov
1The authors are grateful to HS К for support (project 16-11-10200).
Ivanova Olga Yu. — Saint-Petersburg State University of Aerospace Instrumentation, SUAI, St. Petersburg, Russia. e-mail: s.vostokov@sphu.ru
Vostokov Sergei V. — Saint Petersburg State University, St. Petersburg, Russia. e-mail: s.vostokov@sphu.ru,
Zhukov Igor B. — Saint Petersburg State University, St. Petersburg University, 7/9 Univer-sitetskava nab., St. Petersburg, 199034 Russia. e-mail: i.zhukov@spbu.ru
This article links Kurihara's classification of complete discrete valuation fields and Epp's theory of elimination of wild ramification.
For any complete discrete valuation field K with arbitrary residue field of prime characteristic one can define a certain numerical invariant r(K) which underlies Kurihara's classification of such fields into 2 types: the field K is of Type I if and only if r(K) is positive. The value of this invariant indicates how distant is the given field from a standard one, i.e., from a field which is unramified over its constant subfield k which is the maximal subfield with perfect residue field. (Standard 2-dimensional local fields are exactly fields of the form fcjjt}}.)
We prove (under some mild restriction on K) that for a Type I mixed characteristic 2-dimensional local field K there exists an estimate from below for [/ : k] where l/k is an extension such that IK is a standard field (existing due to Epp's theory); the logarithm of this degree can be estimated linearly in terms of r(K) with the coefficient depending only on eK/k.
Keywords: Higher local fields, wild ramification
Bibliography: 15 titles.
For citation:
0. Ivanova, S. Vostokov, I. Zhukov, 2019, "On two approaches to classification of higher local fields" , Chebyshevskii sbornik, vol. 20, no. 2, pp. 186-197.
1. Introduction
In the current paper we develop and compare two approaches to the classification of 2-dimensional local fields in the mixed characteristic case. Here a 2-dimensional local field is a complete discrete valuation field K such that its residue field K has, in its turn, a structure of a complete discrete valuation field with perfect residue field of characteristic p > 0.
If char K = char K, the field K can be identified (non-canonicallv) with the field of formal Laurent series K((X)). However, if char K = 0 and char K = p, there is no explicit description and exhausting classification of such fields K. Here are some known results in this direction.
First of all, there is an important subclass of such fields K, so called standard fields. For any complete discrete valuation field K with the residue field of characteristic p > 0, one can introduce its constant subfield k which is a maximal subfield of K with perfect residue field. It can be proved that in the mixed characteristic case such k is unique. The field K is said to be standard if e^/k = 1 where e^/k is defined in 2.1.
This rather abstract definition working for any complete discrete valuation field with imperfect residue field, takes a very explicit form if K is a 2-dimensional local field. Namely, if K is standard and k is its constant subfield, then
Abstract
oo
conversely, UK = fc{{i}} fa a (one-dimensional) local field k, then K is standard, and k is its constant subfield (see [8] or [14]). Note that in the very classical case, when the residue field of K is finite, k can be constructed as the maximal algebraic extension of Qp inside K.
Obviously, any 2-dimensional local field K with local parameters (n, t) is a finite totally ramified extension of its standard subfield K0 = k{{t}}, where k is the constant subfield of K, and k, t are as in 2.2 A non-trivial result following from Epp's theorem on elimination of wild ramification (see [1], [13]) is that for any such K there exists a constant (i. e. defined over k) finite extension K'/K such that K' is a standard field. In fact, there is a huge freedom in the choice of such K'/K, see [6]. However, the minimal degree dm(K) of such K'/K can be arbitrarily large even in the simplest case [K : Kq\ = p. Thus, dm(K) seems to be an interesting invariant in the classification of 2-dimensional local fields.
Another approach to classification of mixed characteristic complete discrete valuation fields was initiated by Kurihara in [7] to study Milnor K-groups (see [9] or [4]), These groups are applied in class field theory (see [10], [11], [4], [5]). Kurihara subdivides such fields into 2 types. For this, one considers any non-trivial relation a ■ dw + b ■ dt in the module of differentials of the given field K over its constant subfield k, where (w,t) are any local parameters of K. The field K belongs to Type I if vk(a) < vk(b) and to Type II otherwise (see [7], corollary 1.2 and definition 1.3). In particular, all standard fields belong to Type I since k can be chosen from k, and one can take a = 1, b = 0. Kurihara showed that the structure of extensions for the fields of Type I and Type II is very different. For example, K has cyclic wild (resp. ferocious) p-extensions of any degree if and only if K is of Type I (resp. Type II).
A refinement of this classification along with a number of new properties has been given in [2, 3]. It was suggested to consider values like
where tL is a second local parameter in a certain standard field L containing K, and the partial derivatives are used in the usual sense via identification L = l{{tL,}}- It is easy to see that
so, the field K is of Type I if and only if A(n, t) > 0 fa any choice of local parameters t. It can be shown that for the fields of Type I A(^, t) does not depend on the choice of t. For such fields, the value
is an invariant of K measuring resemblance between K and standard fields. In particular, r(K) = if and only if K is "almost standard": a certain unramified extension of K is a standard field.
In this article we obtain a lower bound for dm(K) for a mixed characteristic 2-dimensional local field of Type I, in terms of r(K) and ramification index of the field over its standard subfield. This is accomplished under a certain mild restriction on K (Corollary 5.3.1).
We are grateful to the referee of the first version of this article for valuable remarks.
2. Notation and basic definitions
A(M)
1
(vK(b) - vK(a)),
r(K)= sup A(n,t)
v(w)=1
The following notation is used throughout the paper:
p always denote a prime integer;
vp(x) is the ^adic exponent of an integer number x.
2.1. Discrete valuation fields
For a discrete valuation field F, we denote its valuation bv vf and its residue field by F. For any such F it will be always assumed that charF = p > 0. If char F =p > 0 we put eF = vf(p)-An element kf such that vf (ttf) = 1 is said to be a uniformizer or F.
Denote
Of = {x eF | vf (x) ^ 0};
Uf = {x e F | vf(x) = 0};
Uf(n) = {x e F | vf(x - 1) ^ n} for n e N.
Let L/F be an extension of valuation fields, vl be a valuation on ^^d vl induces the valuation w on F. We denote bv e^/F the index of w(F*) in vl(L*).
A finite extension E/F of discrete valuation fields is said to be
• unramified, if eE/F = ^d E/F is separable;
• tame, if p\ e E/F-, and E/F is separable;
• ferocious, if e E/F = 1, and E/F is purely inseparable;
• totally ramified, if eE/F = |E : F|.
By Vo we denote the valuation on any field normalized so that V0(p) = 1.
L/ F ( L/ F)
of any generator a of Gal(L/K):
s(L/F) = inf vL(a(x)x-1 - 1).
2.2. Two-dimensional local fields
Let K be a two-dimensional local field; denote by
K « = K its first residue field, and by K(0) = K(1) its last residue field, ft is always assumed in this article that charK = 0, char K = p >
0, and K(0) is perfect.
Any two-dimensional mixed-characteristic local field K satisfies the conditions of 2.1. We will use the same notation, that is ex = vk(p), Ok = {x e K | vk(x) ^ 0^d v0 is such that 0( ) = 1
For the valuation of rank 2 on K we use notation vk = (V^, vk): K ^ Z2; tere Z2 is linearly ordered as follows: (a, b) < ( c, d), if b < d or b = d and a < c.
K (0) is a perfect subfield in K (i1) = ^or d e K(0), its Teichmiiller representative in Ok is well defined. We denote it by [ &].
Given Vk, we can define local parameters: a uniformizer k with Vk(k) = (0,1), and a "second local parameter'^ with Vk(t) = (1,0).
The constant subfield of K is its maximal subfield such that its residue field (with respect to vk) is perfect. In particular, if the last residue field of K is finite, the constant subfield of K is the algebraic closure of Qp in K.
In what follows K denotes always a two-dimensional local field, and k is its constant subfield.
The field K is said to be standard, if eK/k = 1-
A finite extension L/K is said to be constant if L = IK where I is an algebraic extension of k.
2.3. Kurihara's classification and related invariants
Let K0 = k{{t}} be a standard 2-dimensional field. For x e K0 its formal derivative || is defined as follows. If x = Y1 ait1 wit h ai eh, then
I =
It is easy to see that || is a well defined element of K0.
Let K0 and L0 be standard fields with K0 C L0, and 1 et t, t' be second local parameters of these fields. Then
dx dx dt dV = ~dt~dtr
where the first factor in the right hand side is the image in L0 of the respective element of K0. Let K0 be a standard field, t a second local parameter of K0, and a,b e K*. Introduce
C(a,b)= V0(- V0(- (a)+ Vo(b)-
Now we check that c(a, b) is independent of the choice of K0 and the second local parameter t0. Let Ki and K2 be standard fields with the second local parameters t\ and t2, and let c\(a,b) and c2(a, b) be functions corresponding to these fields. There exists a standard field E containing both and K2. Let tE be any second local parameter of E. We have
( dx \ ( dx \ ( dti \
v°(m) = + "»(«l)- ' = 1'2;
therefore,
Cl(a, b) - c2(a, b) = V" (|-) - () - (g-) + (^) = ( da \ / db \ / da \ / db \
= Hdt~E) —Hd^) —Hdt~E] + Hd^) = 0
Note that for any x,y,z we have
c(x, y) = c(x, z) — c(y, z), c(x, y) = -c(y, x).
In [2, 3] the notation Ak(ft, t) was used for vk (dftt^) —vk (dtt^, where ft, t are local parameters of K, and tL is a second local parameter of a standard field L which is a finite extension of K. In this article we redefine Ak (ft, t) using v0 instead of vk, i- e.,
A(ft, t) = AK(ft, t) = v^dfttL) — v^dtt^j.
It is shown in [7, 2] that if the condition Ak(ft,t) > 0 is satisfied for some local parameters ft and i of K, then it is satisfied for any pair of local parameters. A field K is of Type I if this condition is satisfied and K is of Type II otherwise (see [2], proposition 4.3). For a field of Type I, A(ft, t) is independent of the choice of the second local parameter t (see [2], Cor. 4.4); its value will be denoted by Ak (ft)- Note that
Ak (ft, t) = c(ft, t) + V0(ft) — V0(t) = c(ft, t) + —.
eK
For a field K of Type I, denote
r(K) = max(AK(ft)\ft e K*, vK(ft) = 1),
rc(K) = max(AK(ft)\ft e K*, vK(ft) = 1) — —.
eK
Then for any second local parameter t of K we have
rc(K) = max(c(ft,t)\ft e K*, vK(ft) = 1).
3. Properties of c(a)
3.1 Proposition. Let a,b e K. Then:
1. min{c(ab, a), c(ab, b)} ^ 0.
2. c(a-1, a) = 0.
3. c(ap, a) = 1.
Proof. Direct calculation.
3.2 Lemma. Let K = k{{t}} be standard, and let nk be a uniform,izing element of k. Then any a e K can be represented (non-uniquely) as
N
a = flM + U, (1)
r
œ + 7 nk jr
r=°
where aœ G k, N ^ 0 a, G Z, and for each r either fr = 0 or
fr = ¿21° tP'1 i£Z
9ri G exists i such that 9r>i = 0 p\i.
For any such representation we have
Proof. See [2, Lemma 4.5].
3.3 Proposition. Let K be of Type I. Let a G Ok; assume
a = 7mf mod7m+10K, (2)
f = EiW,
i£Z
0i G Kexists i such that di = 0. Then
min{c(nmf,a), c(7mf,n)} ^ 0. Proof. It is sufficient to prove that
c(7mf, t) ^ min(c(a, t), c(n, t)). Let L be a standard field, L D K, and let tL be a second local parameter of F For any x G L let
d(x) = dxtl^J .
We have
d(a) = c(a, t) + v°(a) — d(t) ^ M + me^1 — d(t)
with M = min(c(a, t), c(k, t)). Note that the value of d for each term in the expansion (1) for a cannot be less than d(a)\ it follows
7~LmeL/Ka G k((tpM-d(t))). (3)
In particular,
ftT^ft e k((TM-m)). (4)
Let
r = min{vp(i) \ di = 0}. Combining (2), (3) and (4), we conclude that r ^ M — d(t). Terefore,
c( f, t) = c( f, tL) + d(t) ^ r + d(t) ^ M.
Applying Lemma 3.1, we obtain
c(ftmf, t) ^ min(c(ft, t), c(f, t)) ^ M.
Let us say that f e k{{T}} is normalized if either f e Uk, or f e Ok{{T}} and f e k((Tp)). Let ft, t be any local parameters of K.
3.3.1 Corollary. Let u e Uk, where K is of Type I. Then
u = n(i+ftimpn*)),
where for any i either fi = 0 or fi is normalized and Hi ^ 0, and for any such representation we have
min{c(1+ft%U(tpni),u), c(1 + ft*fi(tpni),ft)} > 0 Proof. This follows from Propositions 3.3 and 3.1 by induction.
4. Behavior of c(a) in field extensions
4.1 Lemma. Let K'/K he a finite extension of 2-dimensional local fields, and let X\, x2 e K' he conjugate over K. Then c(x\,x2) = 0.
Proof. Let Li/K(x\) ^e a finite extension such that Li = Ii{{ii}} is a standard field. Then there exists a field L2 D K(x2) and an isomorphism t : L\ ^ L2 over K such that t(x\) = x2.
The field h is exactly the set of elements of Li algebraic over k. ^teefore, l2 = r(h) is the constant subfield of L2.
For any 2 e Li, we have vl2(t(z)) = vl1 (z), since for any L/K the valuation vk has a unique extension to L. ^teefore, = e-L2-, eL2/i2 = eL1/i1 = 1 L2 is standard, and t2 = r(t\) is a second local parameter of L2.
Next, for any 2 e Li it follows from t(1\) = l2 and r(t\) = t2 that
/ dz \ d(t(z)) T\dh) = dt 2 ,
and
_ (dz \ _ fd(r(z)) \
VlA ^J = VL>{ -mT).
Since = &l2 , the same relation is true for v0 instead of vl1 and vl2- Applying this to z = x\, we obtain c(x\,x2) = 0.
4.1.1 Corollary. Let, K'/K he a finite Galois extension. Then the for any x e K' we have
C(Nk'/k(x),x) ^ 0.
Proof. This follows from Lemma 4.1 and Proposition 3.1.
4.2 Lemma. Let K'/K he a finite totally ramified Galois extension, and let K' he of Type I. Then K is of Type I, and TC(K') < TC(K).
K
Set s = rc(K'), if rc( K') is finite, and denote an arbitrary number by s otherwise. We claim that rc(K) ^ s. Let t' be a common second local parameter of K and K'. Let 7k' be a uniformizer of K' such that c(kk',t') ^ s. Then 7k = NK'/k(kk') is a uniformizer of K. Applying Corollary 4.1.1, we obtain
rc(K ) ^ c(7K ,tr) = c(7K' ,t') +c(7K ,7K' ) = S + c(NK'/K (7K' ),7r ' ) ^ S.
4.3 Lemma. Let, K'/K he a tame extension. Then K and K' are of the same type, and, if they are of Type I, then rc(K') = TC(K).
rank
Our definition of r(K) is tailored for fields of Type I only, and we do not know how a parallel result for Type II case can look like.
Proof. The fields K and K' are of the same type by [7, Corollary 1.6].
Assume they are of Type I. Let M/K be the maximal unramified subextension in K'/K. Then M/K' is totally ramified. We will prove that r(M) = r(K), r(K') = r(M). It is sufficient to check the inequalities:
rc(K) < rc(M) < rc(K') < rc(M) < rc(K).
Denote by tx and îm arbitrary second local parameters of K and M. Then îm is also a second K
c(tK, tM) = 0. (5)
Let L be any standard field containing M and tL be its second local parameter. The extension M/K is separable; therefore,
tx = ai?M,
where ai G M, and there exists i such that p\i, a = 0. It follows
and so c(tk, tM) = c(tk, tl) - c(tm, tl) = 0.
1) We prove rc(K) ^ rc(M^. ^rnote s = rc(K) if rc(K) is finite, and let s be arbitrary otherwise.
Let kk be a uniformizer of K such th at c(ttk , tK) ^ s. Then kk is also a uniformi zer of M. Using 5 we obtain
rc(M) ^ c(ttk, tM) = c(ttk, tk) ^ s.
2) Now we prove rc(M) ^ rc(K'), rc( M) ^ rc(K). In view of 5, it is sufficient to prove that for any uniformizer nM of M there exist uniformizers nK and nK^f K and K' such that c(ttk, nM) ^ 0 and c(ttk',nM) ^ ^^et E ^e either K or K'. Denote
|M : K|, E = K IK' : M|, E = Kr
I NM/KftM, E = K X =\ftM, E = Kr
In both cases we have x e E and v0(x) = qeE-
Let ftE,i, tE be arbitrary local parameters of E, and let 0 e E V, si e Z u e Ue(1) be such that ftqE i = [d] t^ux. Denote bv t a second local parameter of any standard field which is a finite extension of K', and denote by r any integer number with v0(si — rq) ^ v0(; put s = si — rq. We will prove that the uniformizer
ftE = t~Er u~i/qftE,i is appropriate. Since ftqE = [d]tsEx, we have
i dftE d(ft%) d{[0]t%x) ,s-idtE + dx
qftE ~m = = = [d]stE x~dt +[0]tEft.
Taking into account
we obtain
[ 8]*tE1^) > V0(S) > V0(f), /dftE\ , s fdx\
V0{^rJ+ (q — 1)eE > V0Vm)
and
c(ftE, x) = (V0^-ftf) — V0(ftE^ — (V0^— V0(x)^ ^ 0.
In the case E = K' we obtained the desired inequality, whereas in the case E = K it is a consequence of the above formula and Corollary 4.1.1.
3) It remains to prove rc(K'') ^ rc(M). This follows from Lemma 4.2.
5. Estimate
We generalize the notion of "being not in touch "introduced in [15] in the prime characteristic case. Let Li/F and L2/F be totally ramified Galois extensions of degree p, and denote si = s(Li/F), s2 = s(L2/F). The extensions Li/F and L2/F are said to be not in touch if either si = s2 or s(L/F) = si = s2 fa any subextension L/F in LiL2/F of degree p.
Next, finite totally ramified Galois p-extensions Li/F and L2/F are not in touch, if for any intermediate fields F C Si C T c where Si/F is normal and Ti/Si is a Galois extension of degree p (i = 1,2), the extensions TiS2/SiS2 and SiT2/SiS2 are not in touch.
The idea behind this notion is that we consider extensions "in general position"such that the ramification of their compositum can be computed in terms of ramification of the original extensions, compare [12, 4.3].
We say that an extension K'/K is constant free, if K'/K is not in touch with any constant extension of K. For example, for K = k{{t}}, where k contains a primitive pth root of unity, a Kummer extension ^ 1 + ft^a)/K with a e Uk is constant free ifia/k.
5.1 Lemma. Let, Li/K and L2/K he Galois extensions of degree p that are not in touch. Assume that LiL2/K is totally ramified. Then:
1. s(LiL2/L2) > s(Li/K);
2. Ifs(Li/K) = p-j, then s(L1L2/L2) > s(Li/K).
Proof. The first part follows immediately from Lemma 3.3.1 in [12]. (It is assumed there that the residue field is perfect but the proof goes through assuming only that LiL2/K is totally ramified.)
For the second part it is sufficient to notice that s(L2/K) < 2—-. Indeed, if
s( Li/K ) = s(L2/K ) =
p-
peK
-1
and L1L2/K is totally ramified, then L1/K and L2/K are always in touch; this can be seen from
5.2 Lemma. Let K contain a primitive pth root of unity. Let K1/K be a totally ramified extension of degree p; denote by n1 any uniform,izer of K1. Let u e K be such that K(tfu) is not in touch with K1 /K and either vK(u) = 1 or 0 < vK(u — 1) < p \ vK(u — 1). Assume that c(u, tL) ^ N and c(n1, tl) ^ N for some integer N ^ 3 and for some standard field L = l{{tL}} containing K1. Then u = u1 If, where u1,b e K1 are such that p \ vk1 (u1 — 1) and c(u1, tL) ^ N — 1.
Proof. Let c(x) = c(x, tL) and let i0 = vK(u — 1). By Corollary 3.3.1
u = <K(«) ^ (1+^/^))
i^pio
with fa normalized or fa = 0, and c(1 + 7tlfi(tpnz)) ^ N. ft follows for fa = 0 that
c(1 + n\fi(tpH))
and by Lemma 3.2
n = c(fi(tp"*)) ^ min(c(m), c(7rifi(tpn>))) ^N — ieKl.
We conclude that n > 0 for i < NeKl.
Denote i1 = min{f : fa = 0, p \ i}\ we have i1 ^ peKl/(p — 1) < NeKl since N ^ 3. Introduce
____1 | „Pi f (4-pn'Pi \
u1 = U(1+7\fi(tPni)) X TT \ l-—
1 iifi( 1 )) i (1+ni/pT1 (^-1))p
= (1+^1)(1 + ^2).
Here $ denotes application of Frobenius automorphism to the coefficients of a power series from k{{T }}.
We SGG from Lemma 5.1 and [12] that
V(u1 — 1) < ^ — S(K1(^¡u1)/K1) p — 1
< p^Kr — S(K () (6)
p — 1
peK-! i pek p — 1 \p — 1
e k-1 + i 0.
Since vKl(S1) = ¿^d vKl(S2) = eKl + i0, we obtain that the initial terms in S1 and S2 do not cancel, whence vKl(S1 + S2) is exactly min(vKl (S1), vKl(S2)). If io = 0 this gives p \ v(u1 — 1), since both i1 and eKl + i0 are not divisible by p. If i0 = 0, we still have p \ v(u1 — 1). Indeed, in this case we have a strict inequality in (6) by the second part of Lemma 5.1, whence v(u1 — 1) = i1.
u = u1 p
b = 7\K (U) n (1+-7i/PT (t^-1 ))-1.
It remains to estimate c(l + ^f^ (tp pz )). Denoting by g, by definition we have
c(l + n\g(t^-)) ^ mm(vo(K\-1g(t^^tL),
vo(pn^-\\g '(t ^ )t^ dt tL)) ^ min(iek\ + c(ni), (npi - 1) + ie^1 + c(t)) ^ min(îe ~K\ + N,N - pie~K\ - 1 + ie~K\ + c(t)) > N - 2.
It follows c(b) ^ N - 2, and c(m) ^ min(c(u), c(b-p)) ^ N - 1.
5.3 Proposition. Let K be of Type I, not almost standard, with TC(K) > n+3, where n = vp(eK/k)■ Assume that K/k{{t}} is constant free (for some choice oft). Let K'/K he a constant extension of degree p. Then K' is of Type I, not almost standard, and TC(K') ^ TC(K) - n - 3.
K = k K k / k p
k / k K / K
have K' = K(c0), where = a <E k*. Consider a chain of subfields
K0 CKi C---CKn = K,
where each Ki+i/Ki is totally ramified of degree p, and K0/k{{t}} is tame. Denote by L any standard field containing K'\ put c(x) = cl(x, t£)-
Let ftn = ft be a uniformizer of K with c(ft) = N, where N = rc(K), and let fti — NK/K.ft-, we have c(fti) ^ N, 0 ^ i ^ n (by Corollary 4.1.1).
Applying Lemma 5.2 to Ki/K0^.., Kn/Kn-i, we obtain that K' = K(x), xp = u e K, p \ v(u — 1) c(u) ^ N — n. In particular, K'/K is totally ramified, whence K' is not almost standard. We have
c(x — 1) = c(x) + v0(x) — v0(x — 1) =
= c(u) — 1 — V0(x — 1) ^ N — n — 1--^ N — n — 3.
p — 1
Pick integers i and j such that Vk' (ft') = 1, where ft' = (x — 1)%ftK Then
c(ft'') ^ min( — 1), c(ft)) ^ N — n — 3, and this proves that K' is of Type I with
rc(K') ^ c(ft') ^ rc(K) — n — 3.
5.3.1 Corollary. Let K be as in Proposition 5.3, with r(K) > m(n + 3), where n = vp(e^/k)> m a positive integer. Assume that l/k is an extension such that IK is almost standard. Then the inequality [I : k] ^ pm holds.
Afterword
Thus, we have established, under some restrictions, a relation between two invariants measuring how far is a given 2-dimensional local field from being standard. We expect that this relation, in some refined form, can be extended to all higher local fields.
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Получено 26.08.2018 г.
Принято в печать 12.07.2019 г.
