Progress in Mathematics Vol. 48 Edited by J. Coates and S. Helgason Birkhauser Verlag Boston . Basel . Stuttgart A. Frohlich Classgroups and Herlrlitian Modules 1984 Birkhauser Boston . Basel . Stuttgart Author: A. Frohlich Mathematics Department and Mathematics Department Imperial College Robinson College London Cambridge England England Library of Congress Cataloging in Publication Data Frohlich, A. (Albrecht), 1916- Classgroups and Hermitian modules. (Progress in mathematics ; vol. 48) Bibliography: p. Includes index. 1. Class groups (Mathematics) 2. Modules (Algebra) 1. Title. II. Series: Progress in mathematics (Boston, Mass.) ; vol.48. QA247.F7583 1984 512'.74 84-11109 ISBN-13: 978-1-4684-6742-0 CIP-Kurztitelaufnahme der Deutschen Bibliothek Frohlich, Albrecht: Classgroups and hermitian modules I A. Frohlich. - Boston; Basel; Stuttgart: Birkhiiuser, 1984. (Progress in mathematics; Vol. 48) ISBN-13: 978-1-4684-6742-0 NE:GT All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. © Birkhiiuser Boston, Inc., 1984 Sof'tcover reprint of the hardcover 1st edition 1984 ISBN-13: 978-1-4684-6742-0 e-ISBN-13: 978-1-4684-6740-6 DOl: 10.1007/978-1-4684-6740-6 9 8 7 6 5 4 3 2 I To Ruth, Sorrel and Shaun An earlier version of these notes has been circulated and quoted under the title "Classgroups, in particular Hermitian Classgroups" VII PREFACE These notes are an expanded and updated version of a course of lectures which I gave at King's College London during the summer term 1979. The main topic is the Hermitian classgroup of orders, and in particular of group rings. Most of this work is published here for the first time. The primary motivation came from the connection with the Galois module structure of rings of algebraic integers. The principal aim was to lay the theoretical basis for attacking what may be called the "converse problem" of Galois module structure theory: to express the symplectic local and global root numbers and conductors as algebraic invariants. A previous edition of these notes was circulated privately among a few collaborators. Based on this, and following a partial solution of the problem by the author, Ph. Cassou-Nogues and M. Taylor succeeded in obtaining a complete solution. In a different direction J. Ritter published a paper, answering certain character theoretic questions raised in the earlier version. I myself disapprove of "secret circulation", but the pressure of other work led to a delay in publication; I hope this volume will make amends. One advantage of the delay is that the relevant recent work can be included. In a sense this is a companion volume to my recent Springer-Ergebnisse-Bericht, where the Hermitian theory was not dealt with. Our approach is via "Hom-groups", analogous to that followed in recent work on locally free classgroups. In fact our notes also in clude the first really systematic and comprehensive account of this approach to classgroups in general. Moreover, the theory of the Hermitian classgroup has some new arithmetic features of independent interest in themselves, and one of our aims was to elaborate on these. I want to record my thanks to all those involved in the new Mathe matics Institute in Augsburg, who took over so willingly and effi ciently the physical production of these notes. IX TABLE OF CONTENTS Page Introduction xi Chapter I Preliminaries §1 Locally free modules and locally freely presented torsion modules §2 Determinants and the Hom language for 5 classgroups §3 Supplement at infinity 18 Chapter I! Involution algebras and the Hermitian classgroup 20 § 1 Involution algebras and duality 20 §2 Hermitian modules 25 §3 Pfaffians of matrices 27 §4 Pfaffians of algebras 33 §5 Discriminants and the Hermitian classgroup 43 §6 Some homomorphisms 56 §7 Pulling back discriminants 69 §8 Unimodular modules 72 §8 Products 76 Chapter II! Indecomposable involution algebras 78 §1 Dictionary 78 §2 The map! 84 §3 Discriminants once more 94 §4 Norms of automorphisms 103 §5 Unimodular classes once more 106 x Page Chapter IV Change of order 117 § 1 Going up 117 §2 Going down 124 Chapter V Groups 146 § 1 Characters 146 §2 Character action. Ordinary theory 155 §3 Character action. Hermitian theory 163 §4 Special formulae 173 §5 Special properties of the group ring 177 involution §6 Some Frobenius modules 178 §7 Some subgroups of the adelic Hermitian 189 classgroup Chapter VI Applications in arithmetic 198 §1 Local theory 199 §2 The global discriminant 215 Literature 221 List of Theorems 224 Some further notation 225 Index 226 XI INTRODUCTION The original motivation for the theory described in these notes stems from the study of "Hermitian modules" over integral group rings, and more generally over orders. The forms considered are more general than those on which the main interest of topologists and K-theorists had been focused, in that now no condition of non-singularity in terms of the order (rather than the algebra) is attached. The significance of such more general forms comes in the first place from algebraic number theory: the ring of integers in a normal extension is a Galois module with an invariant form, in terms of the trace. Topologists have however also had to consider such forms. Apart from this application, our results are of independent arith metic interest in that they generalise classical ones on quadratic or Hermitian lattices. The central theme is the "discriminant problem" which we shall discuss in some detail later in this introduction, and the central concept for its solution is the Hermitian classgroup. Here, as already in preceding work in a purely module theoretic context (cf. [F7]) we work with locally free modules rather than with projec tives, and classgroups are consistently described in terms of "Hom-groups", i. e., of groups of Galois homomorphisms, also to be dis cussed in some further remarks later in this introduction. It will then become worthwhile, and even unavoidable, to look systematically also at those other classgroups, which are defined before Hermitian structure is introduced, from this new general point of view and within this new convenient language. The reader whose interest is re stricted to these pre-Hermitian aspects should read Chapter I and the relevant parts of Chapters IV and V. The approach to classgroups which we are developing arose out of the investigation of the Galois module structure of algebraic integer rings in tame normal extensions and its connection with the functional equation of the Artin L-function (cf. [F7]; [F12]). XII Subsequently a parallel theory came into being, in the first place in the local context, in which the Hermitian module structure became the principal object of study (cf. [F8], [FlO]); here the form comes from a relative trace, more conveniently expressed however as a form into the appropriate group ring. Again the crux of the theory lies in the connections with (now) local root numbers and Galois Gauss sums. Both in the global case and in the local Hermitian case the crucial link between the arithmetic constants on the one hand and the classgroup invariants on the other is formed by the generalised resolvents, and it is at this stage that the new Hom language for classgroups becomes absolutely vital. Let then F be a field (say of characteristic zero), F its c algebraic closure, S'2F = Gal(Fc/F) the absolute Galois group over F. Let r be a finite group - which turns up as relative Galois group over F - and Rr the additive group of virtual characters. Then the various classgroups are to be described in terms of groups such as Ho~ (Rr, G) for varying S'2F-modules G • Resolvents give rise to F elements of such groups, or of related ones. On the other hand if e.g., F is a numberfield with ring of integers 0 then the class group Cl(or) of the integral group ring or appears as a·quotient of J the idele group. The class in Cl(or) of the ring of integers in a tame, normal extension of F, with Galois group r is then described in terms of the above Hom-group via the resolvents. Going beyond group rings, if we consider orders in a semisimple algebra A over F, we have to replace Rr by a corre sponding object, the Grothendieck group KA,F of (equivalence classes of) matrix representations of A over Fc' i.e., we study Hom groups Ho~ (KA F' G). In this language the determinant of a F ' matrix, or more generally the reduced norm of an element a of * A = GLI (A) or of GL n (A) is replaced by a Galois homomorphism * KA,F .... Fc (multiplicative group), again called the determinant and denoted by Det(a). It maps the representation class X (in the group case the character X) given by a matrix T into the determinant Detx (a) = Det T(a) E Fc* Even apart from its suitability for the arithmetic applications, the advantages of a consistent use of the Hom language are tremendous.