Graduate Texts in Mathematics S Axler 67 Editorial Board F.W Gehring K.A Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 T AKEUTI/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nd ed HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTIIZARING Axiomatic Set Theory HuMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CoNWAY Functions of One Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules 2nd ed GOLUBITSKY/GuiLLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKJISAMUEL Commutative Algebra Vol.! 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67) Translation of Corps Locaux Bibliography: p lncludes index Class field theory Homology theory I Title II Series QA247.S4613 512'.74 79-12643 L' edition originale a ete publiee en France sous le titre Corps locaux par HERMANN, editeurs des sciences et des arts, Paris AH rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC © 1979 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Berlin Heidelberg in 1979 Softcover reprint of the hardcover 1st edition 1979 765 ISBN 978-1-4757-5675-3 ISBN 978-1-4757-5673-9 (eBook) DOI 10.1007/978-1-4757-5673-9 SPIN 10761519 Contents Introduction Leitfaden Part One LOCAL FIELDS (BASIC FACTS) Chapter I Discrete Valuation Rings and Dedekind Domains §1 §2 §3 §4 §5 §6 §7 §8 Definition of Discrete Valuation Ring Characterisations of Discrete Valuation Rings Dedekind Domains Extensions The Norm and Inclusion Homomorphisms Example: Simple Extensions Galois Extensions Frobenius Substitution 13 15 17 20 23 Chapter II Completion 26 §1 §2 §3 §4 26 Absolute Values and the Topology Defined by a Discrete Valuation Extensions of a Complete Field Extension and Completion Structure of Complete Discrete Valuation Rings I: Equal Characteristic Case §5 Structure of Complete Discrete Valuation Rings II: Unequal Characteristic Case §6 Witt Vectors 28 30 32 36 40 v Contents Vl Part Two RAMIFICATION Chapter III Discriminant and Different Lattices Discriminant of a Lattice with Respect to a Bilinear Form Discriminant and Different of a Separable Extension Elementary Properties of the Different and Discriminant Unramified Extensions Computation of Different and Discriminant §7 A Differential Characterisation of the Different §1 §2 §3 §4 §5 §6 47 47 48 50 51 53 55 59 Chapter IV Ramification Groups §1 §2 §3 §4 Definition of the Ramification Groups; First Properties The Quotients GjGi+ ~> i The Functions rjJ and ljJ; Herbrand's Theorem Example: Cyclotomic Extensions of the Field QP 61 61 65 73 77 Chapter V The Norm Lemmas The U nramified Case The Cyclic of Prime Order Totally Ramified Case Extension of the Residue Field in a Totally Ramified Extension Multiplicative Polynomials and Additive Polynomials The Galois Totally Ramified Case §7 Application: Proof of the Hasse-Arf Theorem §1 §2 §3 §4 §5 §6 80 80 81 83 87 90 91 93 Chapter VI Artin Representation §1 §2 §3 §4 Representations and Characters Artin Representation Globalisation Artin Representation and Homology (for Algebraic Curves) 97 97 99 103 105 Part Three GROUP COHOMOLOGY Chapter VII 109 Basic Facts §1 G-Modules §2 Cohomology §3 Computing the Cohomology via Cochains 109 Ill 112 Contents §4 §5 §6 §7 §8 Homology Change of Group An Exact Sequence Subgroups of Finite Index Transfer Vll 114 115 117 118 120 Appendix Non-abelian Cohomology 123 Chapter VIII Cohomology of Finite Groups 127 §1 §2 §3 §4 §5 127 129 131 132 134 The Tate Cohomology Groups Restriction and Corestriction Cup Products Cohomology of Finite Cyclic Groups Herbrand Quotient Herbrand Quotient in the Cyclic of Prime Order Case Chapter IX Theorems of Tate and Nakayama 138 §1 §2 §3 §4 §5 138 139 141 142 144 p-Groups Sylow Subgroups Induced Modules; Cohomologically Trivial Modules Cohomology of a p-Group Cohomology of a Finite Group ~D~R~~ 1~ §7 Comparison Theorem §8 The Theorem of Tate and Nakayama 147 148 Chapter X Galois Cohomology 150 §1 §2 §3 §4 §5 §6 §7 150 152 154 155 157 160 161 First Examples Several Examples of "Descent" Infinite Galois Extensions The Brauer Group Comparison with the Classical Definition of the Brauer Group Geometric Interpretation of the Brauer Group: Severi-Brauer Varieties Examples of Brauer Groups Chapter XI Class Formations 164 §1 §2 §3 §4 §5 164 166 168 171 173 The Notion of Formation Class Formations Fundamental Classes and Reciprocity Isomorphism Abelian Extensions and Norm Groups The Existence Theorem Appendix Computations of Cup Products 176 viii Contents Part Four LOCAL CLASS FIELD THEORY Chapter XII Brauer Group of a Local Field 181 §1 Existence of an Unramified Splitting Field §2 Existence of an Unramified Splitting Field (Direct ProoO §3 Determination of the Brauer Group 181 182 184 Chapter XIII Local Class Field Theory 188 §1 §2 §3 §4 §5 188 190 192 195 199 The Group Z and Its Cohomology Quasi-Finite Fields The Brauer Group Class Formation Dwork's Theorem Chapter XIV Local Symbols and Existence Theorem §1 §2 §3 §4 §5 §6 §7 General Definition of Local Symbols The Symbol (a, b) Computation of the Symbol (a, b)v in the Tamely Ramified Case Computation of the Symbol (a,b)v for the Field QP (n = 2) Thesymbols[a,b) The Existence Theorem Example: The Maximal Abelian Extension of QP 204 204 205 209 211 214 218 220 Appendix The Global Case (Statement of Results) 221 Chapter XV Ramification 223 §1 Kernel and Cokernel of an Additive (resp Multiplicative) Polynomial §2 The Norm Groups §3 Explicit Computations 223 226 229 Bibliography 232 Supplementary Bibliography for the English Edition 235 Index 239 Introduction The goal of this book is to present local class field theory from the cohomological point of view, following the method inaugurated by Hochschild and developed by Artin-Tate This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field For example, such fields are obtained by completing an algebraic number field; that is one of the aspects of "localisation" The chapters are grouped in "parts" There are three preliminary parts: the first two on the general theory of local fields, the third on group cohomology Local class field theory, strictly speaking, does not appear until the fourth part Here is a more precise outline of the contents of these four parts: The first contains basic definitions and results on discrete valuation rings, Dedekind domains (which are their "globalisation") and the completion process The prerequisite for this part is a knowledge of elementary notions of algebra and topology, which may be found for instance in Bourbaki The second part is concerned with ramification phenomena (different, discriminant, ramification groups, Artin representation) Just as in the first part, no assumptions are made here about the residue fields It is in this setting that the "norm" map is studied; I have expressed the results in terms of "additive polynomials" and of "multiplicative polynomials", since using the language of algebraic geometry would have led me too far astray The third part (group cohomology) is more of a summary-and an incomplete one at that-than a systematic presentation, which would have filled an entire volume by itself In the two first chapters, I not give complete proofs, but refer the reader to the work ofCartan-Eilenberg [13] as well as to Grothendieck's 'Tohoku" [26] The next two chapters (theorem of TateNakayama, Galois cohomology) are developed specifically for arithmetic 229 §3 Explicit Computations §3 Explicit Computations We return to the situation of th Thus L/K is a finite, totally ramified abelian extension, with Galois group G We have seen (cor to th 1) that the reciprocity isomorphism w:K*/NL* ~ G transforms the subgroups u;uNut ofK */NL *into the ramification groups G" of G Passing to the quotient, we get isomorphisms w": UK./UK.+ NUt such that Ny = x (which is possible by Chap V, §6, cor to prop 9) Put z = yF- ; then z = ns- z', with s E G" and z' E uc>+ and = bn(a) =s mod G" + We now distinguish two cases: Case a) K has positive characteristic By prop 15 of Chap XIII, there exists y' E Ut">+ such that z' = y'F- • = ns- • If x' = Ny', then (xx'- 1 = 1, whence We have (yy'- 1 xx'- E K * and x' E K * Furthermore, Dwork's theorem (Chap XIII, §5, cor to th 2) shows thatw(xx'- 1) = s- As x' belongs to U~ n K* =UK.+ 1, we have w(x') E G"+ and w(x) s- mod G"+ , which proves the proposition in this case t- t- = 230 XV Ramification Case b) K has characteristic zero Then Gn = {1} if n ~ 1, so we can assume n = Moreover, G is cyclic, and if r is its order, the field K contains the group of all rth roots of unity (cf Chap IV, §2) We can choose x andy to be multiplicative representatives (cf Chap II, §4); the element z = yF- is then an rth root of unity, and it is easy to show that it can be put into the form ns- \where n is a uniformizer ofL and sis a suitably chosen element of G We then apply Dwork's theorem as in case a) The preceding proposition reduces the computation of wn to that of C>n, i.e., ultimately to that ofNn We give an example: Proposition Let L/K be a cyclic totally ramified extension of prime degree p equal to the characteristic ofK; let G be its Galois group, and lets be a generator ofG; let t be the largest integer for which G ¥= {1} Let n uniformize L, and set M = s(n)/n - Let x E Uk, and let c(x) = (x - 1)/Tr(M) Then c(x) belongs to the valuation ring of K, and if c(x) denotes its image in K, we have (x, L/K) = SS(c(x)) [For the definition of S: K ~ Z/pZ, see Chap XIV, §4.] Let n' be an element of K such that vK(n') = t Then Tr(M) = bn' and N(M) = an', with a, bE UK-cf Chap V, §3 If we put y = + t]M, with '7 E AL, then N(y) = + (at]P + bry)n' mod v:t \ cf Chap V, §3 The map N is therefore represented by the additive polynomial P1(t]) = ztt]P + Ot], and since N(n - ) = 1, ry = is an element of the kernel of P • According to example of §1, C>p,(~) is equal to-S(o- ~) If x = + ~n', then C> 1(x) = s-m, with m = - S(o- ~) = - c(x), whence theresult follows, taking prop into account Prop can itself be used to compute some local symbols (a, b)v We give only one example: Proposition Suppose that K (resp K) has characteristic zero (resp p), and that K contains the group /Jp of all pth roots of unity Let w be a generator of /Jp· Let e = vK(P) be the absolute ramification index of K, and let t = epj(p- 1), which is an integer If a E K * and b E Uk, then (a b) = ' v wvK(a) · m(b) ' with m(b) = S ( b- ) p(w- 1) · [We write S(c) instead of S(c) if c E AK.J We know from Chap IV, prop 17 that vK(w- 1) = ej(p - 1), which shows that tis indeed an integer On the other hand, the bilinearity of the symbol 231 §3 Explicit Computations (a, b)v allows us to restrict to the case where a is a uniformizer of K Let n be a pth root of a, and let L = K(n); we can apply prop to the extension L/K Choose the generator s of G(L/K) so that ns- = w; we then see that t = epj(p - 1), M = w - 1, Tr(M) = p(w - 1) Hence • ( b- ) (b,L/K)S" (b) , wtth m(b)- S p(w -1) Therefore (a, b)v = n