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Class Formations Ans Higher Dimensional Local Class Field Theory Journal of Number Theory NT2048 journal of number theory 62, 273283 (1997) article no. NT972048 Class Formations and Higher Dimensional Local Class Field Theory Michael Spiess* Mathematisches Institut der Universitat Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany Communicated by P. Roquette Received December 9, 1995 The reciprocity law of higher dimensional local class field theory is proved with View metadata,the help citation of class and formations. similar papers 1997 at core.ac.uk Academic Press brought to you by CORE provided by Elsevier - Publisher Connector 1. INTRODUCTION The purpose of this note is to prove the reciprocity law of higher dimen- sional local class field theory with the help of class formations. We work with generalized class formations as introduced by Koya in [Ko1]. He uses them to prove the reciprocity law for 2-local fields. We extend his method to arbitrary higher dimensional local fields. In order to outline our result we first review higher dimensional local class field theory. Let K be an n-local field, i.e., there is a sequence of fields k0 , k1 , ..., kn=K such that ki is a complete discretely valued field with residue field ki&1 and ab k0 is finite. Let GK be the absolute Galois group of K and GK its maximal M abelian quotient. Let Kn (K) be the nth Milnor K-group of K. The main result of higher dimensional local class field theory is the following Theorem (Reciprocity Law). There exist a canonical homomorphism (the reciprocity map) M ab \: Kn (K) Ä GK with dense image. For each finite abelian extension LÂK, \ induces an isomorphism M M Kn (K)ÂNLÂK Kn (L)$Gal(LÂK). * E-mail: michael.spiessÄmathi.uni-heidelberg.de 273 0022-314XÂ97 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved. File: 641J 204801 . By:CV . Date:06:02:97 . Time:11:13 LOP8M. V8.0. Page 01:01 Codes: 3426 Signs: 1633 . Length: 50 pic 3 pts, 212 mm 274 MICHAEL SPIESS This theorem is due to Kato ([Ka 1, Ka 2]; for an overview of the proof see the survey article of W. Raskind [Ra]). In the case char K>0 it has been proved independently by Parshin [Pa1]. A different and more elementary proof has been found by Fesenko (see [Fe] for an overview of his approach and further references). Kato's proof is based on a duality theorem which is a generalization of the classical Tate duality to higher dimensional local fields. Koya has given a different approach by generaliz- ing the concept of class formations. We are going to recall this briefly. Let G be a profinite group and let C. be a bounded complex of discrete . G-modules whose positive terms are zero. The pair (G, C ) is called a class formation if the following axioms are satisfied:1 (CF1) For all open subgroups H of G, . H 1(H, C )=0. (CF2) For each subgroup H of G there exists an isomorphism 2 . invH : H (H, C ) Ä QÂZ. (CF3) For all pairs of open subgroups VUG the diagram . res . H 2(U, C ) wwÄ H 2(V, C ) invU invV QÂZ m QÂZ commutes with m=[U : V]. A class formation gives rise to an abstract class field theory. This can be summarized as follows . (2.2) Theorem. Let (G, C ) be a class formation. For every open sub- group H there exists a canonical map 0 . ab \H : H (H, C ) Ä H with dense image. If VUG is a pair of open subgroups then the diagram . \V H 0(V, C ) wwÄV ab cor . \u H 0(U, C ) wwÄU ab commutes. If U is normal in V, then \V induces an isomorphism . Coker(cor: H 0(V, C ) Ä H 0(U, C ))$(UÂV)ab. 1 Koya actually uses 4 axioms. However his first axiom seems to be superfluous. File: 641J 204802 . By:CV . Date:06:02:97 . Time:11:13 LOP8M. V8.0. Page 01:01 Codes: 3185 Signs: 1530 . Length: 45 pic 0 pts, 190 mm CLASS FORMATIONS 275 . If C consists of a single G-module in degree 0 this is a classical result . of Tate. If C is two-term complex such that except of the 0th and (&1)th term all terms are zero it has been proved by Koya in [Ko1]. The general case is an easy consequence of the TateNakayamaKoya Theorem [Ko2] as we will see in Section 2. Theorem 2.2 applies to the class field theory of 1- and 2-local fields as follows. If K is a 1-local field then (GK , Gm) is a class formation. If K is a 2-local field then it is shown in [Ko1] (by using (3.12) below) that (GK , Z(2)[2]) is a class formation. Here Z(2) is the 2-term complex intro- duced by Lichtenbaum in [Li2]. Generally for every field F (or more generally for every regular scheme) Lichtenbaum has conjectured the exist- ence of complexes of GF-modules Z(m), m0, satisfying some basic axioms (see [Li1]). The existence of these complexes are known yet only for m2. This is the reason why Koya deals only with the case n=2. By working with the ``decomposable part of motivic cohomology'' Z8 (n) studied by Kahn in [Kn] we are able to give a new proof of the reciprocity law for n-local fields for arbitrary n. In Section 3 we shall show that 8 (GK , Z(n)[n]) is a class formation. As a consequence we obtain the following slightly improved reciprocity law (3.14) Theorem. For every Galois extension LÂK, \ induces an isomorphism M M ab Kn (K )ÂNLÂK Kn (L)$Gal(LÂK) . In the case char K=0, a second proof by using the Bloch complex instead of Z8 (n) will be sketched at the end of section 3. Finally we mention that in the case char K=p>0, Parshin has found a proof of the p-part of the reciprocity law by using class formations in the classical sense (see [Pa2], [Pa3]). Notations. Let G be a profinite group and A a discrete G-module. By G 1(G, A) we denote the fixmodule A . For an integer m, Am and AÂm denote the kernel and cokernel of the map A wÄm A. By D(G) we denote the derived category of the category of discrete . G-modules. For triangels in D(G) we use the notation A Ä B Ä C Ä . A [1]. Let X be a complex of G-modules. By H qX } we denote the qth . + cohomology of X . X [n] denotes the complex (X [n])q=Xn q, d . = . X [n] n . (&1) dX . trX and tr X denote the complexes . r&2 r&1 r tr X : }}} ÄX ÄX ÄKer d Ä 0 . r&1 r r+1 tr X :0ÄIm d ÄX ÄX Ä }}}. If G is a finite group and A is a bounded complex of G-modules then . H q(G, A ) is the qth modified hypercohomology group of G as introduced in [Ko1] and [Kn]. File: 641J 204803 . By:CV . Date:06:02:97 . Time:11:13 LOP8M. V8.0. Page 01:01 Codes: 3277 Signs: 2392 . Length: 45 pic 0 pts, 190 mm 276 MICHAEL SPIESS For a field K let +m be the GK -module of mth roots of unity. We use the j i standard notation ZÂm( j)=+m , j>0 if (char K, m)=1 and ZÂp ( j)= j Wi0X, log if char K=p>0. Finally we set QÂZ( j)= ZÂm( j). I thank the Institute for Advanced Study for hospitality and the Deutsche Forschungsgemeinschaft for support. 2. ABSTRACT CLASS FIELD THEORY In this section we shall prove Theorem (2.2). We need the following (see [Ko2]) (2.1) Theorem (TateNakayamaKoya). Let G be a finite group and . A be a bounded complex of G-modules. Let a be an element of H 2(G, A ). Assume that for every subgroup H of G we have: . (i) H 1(H, A )=0 2 . (ii) H (H, A ) is generated by resGÂH(a) whose order is equal to *(H). Then for every integer q and every subgroup H of G we have . H q&2(H, Z)$H q(H, A ). Now the proof of (2.2) is similar to the proof of [Ko1, Theorem 3.2] so we omit some details. Let H be an open normal subgroup of G. From the triangle in D(GÂH) . t0 R1(H, C ) Ä R1(H, C ) Ä t1 R1(H, C ) Ä t0 R1(H, C )[1] (CF1) and [Ko1, Prop. (1.2)] we get 1 . H (GÂH, t0 R1(H, C ))=0 and 2 . 2 . 2 . H (GÂH, t0 R1(H, C ))=Ker(H (G, C ) Ä H (H, C )). 1 . 1 . Note that H t1 R1(H, C )=H (H, C )=0 hence 2 . 0 2 . 2 . GÂH H (GÂH, t1 R1(H, C ))=H (GÂH, H R1(H, C ))=H (H, C ) . Now (CF2), (CF3) yield 0 . &2 ab H (GÂH, t0 R1(H, C ))=H (GÂH, Z)=(GÂH) . Therefore by [Ko 1, Prop. 1.2] we get an isomorphism . Coker(cor: H 0(H, C ) Ä H 0(G, C ))$(GÂH)ab. File: 641J 204804 . By:CV . Date:06:02:97 . Time:11:13 LOP8M. V8.0. Page 01:01 Codes: 2528 Signs: 1313 . Length: 45 pic 0 pts, 190 mm CLASS FORMATIONS 277 By composing it with the natural projection we obtain a surjective map .
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