FORMAL GROUPS AND APPLICATIONS MICHIEL HAZEWINKEL AMS CHELSEA PUBLISHING

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1 FORMAL GROUPS AND APPLICATIONS MICHIEL HAZEWINKEL AMS CHELSEA PUBLISHING

2 FORMAL GROUPS AND APPLICATIONS

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4 FORMAL GROUPS AND APPLICATIONS MICHIEL HAZEWINKEL AMS CHELSEA PUBLISHING American Mathematical Society Providence, Rhode Island AMERICAN MATHEMATICAL ΑΓΕΩΜΕ ΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ SOCIETY FOUNDED 1888

5 2010 Mathematics Subject Classification. Primary 14L05; Secondary 16Txx, 05Exx, 11Fxx, 11Sxx, 12Fxx, 13F35, 55N22. For additional information and updates on this book, visit Library of Congress Cataloging-in-Publication Data Hazewinkel, Michiel. Formal groups and applications / Michiel Hazewinkel. pages cm Includes bibliographical references and index. ISBN (alk. paper) 1. Formal groups. I. Title. QA177.4.H dc Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island USA. Requests can also be made by to reprint-permission@ams.org. c 1978 held by the author. All rights reserved. Reprinted with corrections by the American Mathematical Society, Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at

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8 Preface to the Corrected Printing Preface Leitfaden and Indicien Introduction xi xiii xv xix

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12 Preface to the corrected printing Some two years ago the idea arose for a second (corrected) printing of my book Formal Groups and Applications, Academic Press, The initiative originated with the American Mathematical Society in the person of Sergei Gelfand. Naturally the plan was to include an update chapter, outlining what had happened since I readily agreed to that, thinking that not all that much had happened (except as regards interrelations of formal groups with algebraic topology) and that some 100 pages would suffice for an update chapter. I was vastly mistaken in my estimates. Meanwhile, I have collected well over 2400 relevant papers in published or preprint form and to do even marginal justice to all this material requires a second volume of Formal Groups and Applications comparable in size to the first one. That second volume, also to be published by the American Mathematical Society, is now in the process of being written. Currently it looks like the manuscript will be ready in the spring of Meanwhile, here is a corrected printing of the original volume from 1978, complete with some three pages worth of corrections and a few short addenda. None of the corrections is very serious and all of the misprints involved can easily be corrected by the (attentive) reader. Still it is probably worthwhile having them listed. There is little hope that I have really caught all misprints; so I will be grateful for the signaling of additional misprints on the part of generous readers. The original edition of Formal Groups and Applications has beenout of print for many years now and seems to be not always easy to get hold of even via university libraries. So I hope and trust that this corrected printing will be useful. Michiel Hazewinkel Bussum, 18 June 2012 xi

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30 théorie des groupes algébriques, CBRM (1962), (E.4.1) M. I. Bašmakov M. I. Bašmakov M. I. Bašmakov A. S. Kuroskin, ˇ M. I. Bašmakov A. S. Kuroskin, ˇ P. Berthelot, Slopes of Frobenius in cristalline cohomology, in R. Hartshorne (ed.), Algebraic geometry, Proc. Symp. Pure Math. 29 (1975), (B.2.3) P. Berthelot, Cohomology cristalline des schémas de caracteristique p>0, LNM 407 (1974). (B.2.3, )

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33 E. J. Ditters, Groupes formels, Cours 3 e cycle 73/74, Univ. Paris, 11th, Orsay, 108p. (1975). (E.5.5, E.5.1, , E.4.3, )

34 J. M. Fontaine, Points d ordre fini d un groupe formel sur une extension non-ramifié dez p, Journées Arithmétiques Grenoble, Bull. Soc. Math. France mém. 37 (1973), (E.5.4) A. Grothendieck, Catégories fibrés et descente, Sém. Géomet. Algéb. 1 (1960/1961), exposé VI. (E.3.1)

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36 M. Hazewinkel, Une théorie de Cartier-Dieudonné pour les A-modules formels, C.R. Acad. Sci. Paris 284 (1977), (E.4.5) N. Jacobson, The theory of rings, Math. Surveys 2, AMS, (35.5.9, E.4.4, ) N. Jacobson, Lectures in Abstract Algebra, 3 vols, van Nostrand-Reinhold, Princeton, New Jersey, 1951, 1953, (35.2.8, )

37 M. Karoubi, Cobordisme et groupes formels, Sém. Bourbaki 1971/72, Exposé 408, Lecture Notes Math. Vol Springer-Verlag, Berlin and New York, 1973.

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42 N. Roby, Lois polynômes et lois formelles en théorie des modules, Ann. Ecole Norm. Sup. 80 (1963),

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69 λ-ring, E.2.1, 17.2 universal, on one generator, E.6.3 Landweber-Novikov operators, Lazard comparison lemma, see comparison lemma Lazard ring, Lie algebra of a formal group (law) D9.7, D14.1, D , T free, D14.4.7, T Lie morphism, 14.3 Lie theory, formal, , 37.4 Lie s third theorem, formal version, 14.5, Lifting (a formal group law), Local-global theorems (for formal group laws), 20.5 Logarithm A, of a formal A module, , of a formal group law, 5.4 of the formal group law of complex cobordism, T31.1.7, T relation with log (1 + z), , Lubin-Tate formal group law (formal A- module) adjointness theorem, T30.2.9, T as functional equation formal group laws, 8.3.6, 13.4 generalized, E9.7, D13.2.1, D more dimensional, 13.3

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