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Ramification Groups of Local Fields with Imperfect Residue Class Fields

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Ramification Groups of Local Fields with Imperfect Residue Class Fields Rami¯cation groups of local ¯elds with imperfect residue class ¯elds version 2/22/2006 Rami¯cation groups of local ¯elds with imperfect residue class ¯elds (To appear in the Journal of Number Theory) Bart de Smit Department of Mathematics University of California Berkeley, CA 94720 Abstract. The Galois group of a Galois extension of local ¯elds with an inseparable residue class ¯eld extension has two intertwined ¯ltrations with rami¯cation groups. This note contains some elementary results on the structure of these ¯ltrations, that are similar to those given by Serre in Corps Locaux, chapter IV, x3, for the case that the residue class ¯eld extension is separable. Key words: rami¯cation groups, local ¯elds. 1980 Mathematics subject classi¯cation (1985): 11S15. Acknowledgement. The author was supported by a John and Elliot Wheeler Fellowship from the University of California and NSF contract DMS 90-02939. 1. Introduction. A local ¯eld is a complete ¯eld with respect to a discrete valuation. It is usually assumed in the theory of Galois extensions of local ¯elds that the residue class ¯eld extension is separable, which holds for completions of number ¯elds and function ¯elds with a perfect ¯eld of constants (see e.g. [7, ch. IV]). We refer to this case as the classical case. There are many examples of imperfect residue class ¯elds though, for instance the residue ¯eld of the local ring of a prime divisor on an algebraic surface in non-zero characteristic. This note is concerned with Galois extensions of local ¯elds with an inseparable residue class ¯eld extension. As indicated in Zariski-Samuel [8], this leads to considering two intertwined ¯ltrations of the Galois group with rami¯cation groups. We will introduce this double ¯ltration in section 2. For the classical case, some basic results on the location of jumps in the sequence of rami¯cation groups can be found in Serre [7, ch. IV, x3]. For instance, it is shown there that the jumps are all congruent modulo the residue characteristic. The goal of this note is to generalize these elementary statements in terms of the double ¯ltration. See Fontaine [4] for an improvement for the case of a perfect residue ¯eld. Deeper results like Hasse-Arf's theorem (see [7]) are even impossible to formulate for inseparable residue ¯eld extensions, because there is no satisfactory de¯nition of an \upper numbering" in this case (see [3] and [5]). The most advanced result for local ¯elds with imperfect residue class ¯elds is probably Kato's local class ¯eld theory in [6] for \n- dimensional local ¯elds" F , in which a canonical homomorphism, the norm residue map, 1 from a certain Milnor K-group to Gal(F ab=F ) is constructed. Hyodo de¯nes rami¯cation groups in the upper numbering by looking at a ¯ltration of this K-group (see [5]), but results on their relation to the usual rami¯cation groups in lower numbering seem to be lacking. 2. Rami¯cation groups. For a local ¯eld K, we denote the valuation ring by AK , its maximal ideal by pK and its residue class ¯eld by K. Let K ½ L be a ¯nite Galois extension of local ¯elds with group G. Assume that char K = p > 0 and let f = fsepfinsep be the degree of L over K, where fsep is the degree of the separable part. Let e = etameewild be the rami¯cation index of L over K, where etame is the tame rami¯cation index (i.e., the p-prime part of e). 2.1 De¯nition. For n; i 2 Z¸0 the (n; i)-rami¯cation group Gn;i of L over K is the subgroup of G = Gal(L=K) consisting of those K-automorphisms of L that induce the i n+i i n+i identity on pL=pL , i.e. Gn;i = f 2 G : 8x 2 pL : x ¡ (x) 2 pL g. Note that Gn;i is a normal subgroup of G, as it is the kernel of the homomorphism G ! i n+i Aut(pL=pL ). For n ¸ 0 we will use the notation Gn = Gn+1;0 and Hn = Gn;1, so that the usual rami¯cation groups in the classical case, as in Serre [7], are the Gn. We have H0 = G, and Gn ½ Hn. G0 The inertia group of L over K is G0. In other words, the ¯eld of invariants T = L is the maximal sub¯eld of L that is unrami¯ed over K, so T is the separable closure of K in L, and G=G0 =» Gal(T =K), which has order fsep. The tame rami¯cation group, corresponding to the maximal tamely rami¯ed sub¯eld of L, is H1 \G0 (see e.g. [2]), which is just H1 by the next proposition. The group G0=H1 is cyclic, as it maps injectively to 2 » ¤ AutL(pL=pL) = L , and its order is etame. Finally, H1 is a p-group of order ewildfinsep. n i i+n n n For n; i 2 Z¸0 we de¯ne Bi = pL=pL , so that B = Li¸0 Bi becomes a graded n AK -algebra with the obvious multiplication map. The action of G on each Bi induces n n an AK -algebra action on B . Note that B is isomorphic to the tensor algebra of the n n B0 -module B1 , and that the isomorphism is canonical (in particular, it preserves the n G-action). An element 2 G lies in Gn;i if and only it acts trivially on Bi . 2.2 Proposition. For all n ¸ 1 and i ¸ 0 we have Hn ½ Gn;i ½ Gn¡1. n Proof. The induced action of an element 2 Gn;i on the ring B0 is compatible with the n n n n trivial action of on the B0 -module Bi . As Bi is a faithful B0 -module it follows that n operates trivially on B0 , so 2 Gn¡1, which proves the inclusion on the right. n Now let 2 Hn, then we see that 2 Gn¡1, and operates trivially on B0 as well n n as on B1 . Therefore, acts trivially on the tensor algebra B , and in particular on the homogeneous part of degree i, so 2 Gn;i. This proves the other inclusion. 2 2.3 Proposition. If n > 1 in the proposition above then we have equality on the left if pi and on the right if p j i. In the case n = 1, the quotient group G1;i=H1 is the i-torsion of G0=H1, which is cyclic of order gcd(etame; i). n n n Proof. Let 2 Gn¡1, so that operates trivially on B0 . The B0 -module B1 is free n n of rank 1, so the B0 -automorphism of B1 induced by is multiplication by a unit u of n i n B0 . Consequently, induces multiplication by u on the i-th tensor power Bi of the n n i i B0 -module B1 , and it follows that 2 Gn;i , u = 1 , 2 Hn. In particular G1;i=H1 is the kernel of multiplication by i on the cyclic group G0=H1. n¡1 n Now suppose that n > 1. As acts trivially on B1 ½ B0 , we must have u 2 n¡1 n n p 1 + pL =pL ½ B0 , so that u = 1. It follows that we have 2 Gn;i if and only if p j i or u = 1, and the ¯rst statement of the proposition follows at once. We thus have a ¯ltration of G as in Zariski-Samuel [8, ch. V, x10, theorem 25]: G = H0 G0 H1 G1 H2 G2 ¢ ¢ ¢ and every rami¯cation group Gn;i with n > 1 occurs in this sequence, in the sense that it equals Gn¡1 or Hn. In the classical case (finsep = 1) we have Hn = Gn for n ¸ 1, so we do not have a strict re¯nement of the usual chain of rami¯cation groups. This can be shown as follows. G0 Let T = L be the inertia ¯eld. Then we have AL = AT + pL as T = L. If 2 Hn n+1 with n ¸ 1, then operates trivially on pL=pL and on AT (as Hn ½ G0), so it must act n+1 trivially on AL=pL which shows that 2 Gn. If ewild = 1, then we have Gn = Hn+1 for n ¸ 1, because multiplication by a prime n n element of the tame rami¯cation ¯eld is an isomorphism of B0 to B1 , that preserves the action of H1. We now derive some simple properties of the rami¯cation groups by embedding the quotient groups Gn=Gn+m and Hn=Hn+m in groups of homomorphisms for m · n. n+1 2.4 Proposition. Let n; m 2 Z with 0 · m · n. The map Á : Gn ! HomAK (AL; pL ) de¯ned by Á()(x) = x ¡ x induces an injective homomorphism of groups m+1 m Gn=Gn+mHomAK (B0 ; Bn+1): m+1 n+m+1 Proof. As Gn ½ Gn;m+1, the image of pL under Á() is contained in pL for m+1 m all 2 Gn, so we obtain a mapping Gn ! HomAK (B0 ; Bn+1). In order to show that this is a homomorphism of groups, note that for ; ¿ 2 Gn and x 2 AL we have 2n+1 n+1 Á(¿)(x) ¡ Á()(x) ¡ Á(¿)(x) = Á()(Á(¿)(x)) 2 pL ; because Á(¿)(x) 2 pL and 2 Gn ½ Gn;n+1. Finally, the kernel of this homomorphism is clearly Gn+m.
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