Cyclic Homology

524 Pages·1998·19.63 MB·English

by  Jean-Louis Loday (auth.)

Grundlehren der
mathematischen Wissenschaften
301
A Series of Comprehensive Studies in Mathematics
Editors
S. S. Chern B. Eckmann P. de la Harpe
H. Hironaka F. Hirzebruch N. Hitchin
1. Hormander M.-A. Knus A. Kupiainen
J. Lannes G. Lebeau M. Ratner D. Serre
Ya.G. Sinai N. J. A. Sloane J.Tits
M. Waldschmidt S. Watanabe
Managing Editors
M. Berger J. Coates S.R.S. Varadhan
Springer-Verlag Berlin Heidelberg GmbH
Jean -Louis Loday
Cyclic
Homology
Second Edition
With 24 Figures
Springer
Jean-Louis Loday
Institut de Recherche Mathematique Av ancee
Centre National de la Recherche Scientifique
7, rue Rene Descartes
F-67084 Strasbourg, France
Library of Congress Cataloging-in-Publication Data
Loday, Jean-Louis.
Cyclic homology I Jean-Louis Loday. - znd ed.
p. cm. - (Grundlehren der mathematischen Wissenschaften; 301)
Includes bibliographical references and index.
ISBN 978-3-642-08316-7 ISBN 978-3-662-11389-9 (eBook)
DOI 10.1007/978-3-662-11389-9
1. Homology theory. I. Tide. II. Series.
QA61Z.3.L63 1998 51Z' .55-dc21 97-22418 CIP
Mathematics Subject Classification (1991):
Primary: 14Fxx, 16E40, 17B56, 18F25, 18Gxx, 19-xx, 46Lxx,
55-xx
Secondary: 05A19, 16E30, 16R30, 16Wxx, 17D99, 20B30, 57R20,
58B30
ISSN 0072-7830
ISBN 978-3-642-08316-7
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© Springer-Verlag Berlin Heidelberg 1992, 1998
OriginaUy published by Springer-Verlag Berlin Heidelberg New York in 1998
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SPIN: 1I7686S4 41/3III-S 4 3 2 Printed on acid-free paper
U ne mathematique bleue
Dans une mer jamais etale
a
D' ou no us remonte peu peu
Cette memoire des etoiles
Leo Ferre
, .-
A Eliane
Preface to the Second Edition
Apart from correction of misprints, inaccuracies and errors, the main differ
ence between the second edition and the first is the addition of a new chapter
on Mac Lane (co)homology, written jointly with Teimuraz Pirashvili. It is
related to Hochschild homology, to algebraic K-theory and cohomology of
small categories as treated in the previous chapters (see the introduction to
Chapter 13). Appendix C has been modified accordingly.
The first list of references was reasonably up to date for papers dealing
with cyclic homology until 1992. It contains all the references mentioned
in Chapters 1 to 12 and in the appendices. Chapter 13 has its own list of
references. Since the publication of the first edition numerous results on the
cyclic theory have appeared, namely about the periodic theory, and also
about topological cyclic homology. For the convenience of the reader we give
a second list of references concerning the cyclic theory for the period 1992-96.
It is a pleasure to thank here all the colleagues who helped me to im
prove this second edition, namely C. Allday, C.-F. Bodigheimer, J. Browkin,
B. Dayton, I. Emmanouil, V. Franjou, A. Frabetti, J. Franke, F. Goichot,
V. Gnedbaye, J.A. Guccione, J.J. Guccione, P. Julg, W. van der Kallen, M.
Karoubi, C. Kassel, B. Keller, M. Khalkhali, J. Lodder, J. Majadas, J. Mc
Cleary, M. Ronco, G. van der Sandt.
It is a pleasure to warmly thank Teimuraz Pirashvili for numerous en
lightening conversations and for his kind collaboration on Chapter 13.
At 48°35'N and 7°48'E, January 21st, 1997.
Preface
II y a maintenant 10 ans que l'homologie cyclique a pris son essor et Ie rythme
a
de parution des publications son sujet confirme son importance. Durant ce
laps de temps l'efIet de sedimentation a pu operer et il devenait possible,
sinon necessaire, de disposer d'un ouvrage de reference sur Ie sujet.
a
Je n'ai pu ecrire ce livre que grace aux enseignements et l'aide de
nombreux collegues, que je voudrais remercier ici. Les cours de topologie
algebrique d'Henri Cartan, qui resteront certainement dans la memoire de
ses auditeurs, ont constitue mon initiation et il est difficile d'en etre digne.
a
Max Karoubi m'a introduit la K-theorie, topologique tout d'abord, puis
a
algebrique ensuite, et son enseignement n'a pas peu contribue rna forma
tion. Dan Quillen a ete constamment present tout au long de ces annees.
Au debut ce fut par ses ecrits (cobordisme et groupes formels, homotopie
rationnelle), puis par ses exposes (K-theorie algebrique) et, plus recemment,
a
par une collaboration qui est l'origine de ce livre. Les conversations et
discussions avec Alain Connes furent toujours stimulantes et exaltantes. Ses
encouragements et son aide furent pour moi un soutien constant. Je voudrais
aussi remercier Zbignew Fiedorowicz, Claudio Procesi et Ronnie Brown pour
a
leur collaboration effie ace et amicale. Remerciements aussi Keith Dennis
pour m'avoir donne l'opportunite de faire un cours sur l'homologie cyclique
a a
Cornell University au tout debut de la redaction et Jean-Luc Brylin
a
ski pour un semestre fructueux passe Penn State University. Ce livre doit
aussi beaucoup ad e nombreux autres collegues, soit pour des discussions, soit
pour des commentaires pertinents, en particulier a L. Avramov, P. Blanc, J.
L. Cathelineau, C. Cuvier, S. Chase, P. Gaucher, F. Goichot, P. Julg, W.
van der Kallen, C. Kassel, P. Ion, J. Lodder, R. MacCarthy, A. Solotar, T.
Pirashvili, C. Weibel et Ie rapporteur. Mamuka Jiblaze a relu entierement Ie
manuscrit durant la phase finale et je lui en sais gre.
Je voudrais aussi mentionner tout particulierement Maria Ronco pour
m'avoir toujours ecoute avec attention, pour avoir lu plusieurs versions de
ce livre et pour avoir corrige de nombreuses imprecisions. Enfin et surtout
je terminerai en remerciant chaleureusement Daniel Guin pour Ie nombre
incalculable d'heures que nous avons passe ensemble devant un tableau noir
et dont je garde Ie meilleur souvenir.
Par 48° 35'N et 7° 48'E, Ie 12 janvier 1992.
Contents
Introduction ................................................ XV
Notation and Terminology .................................. XIX
Chapter 1. Hochschild Homology ........................... 1
1.0 Chain Complexes ......................................... 2
1.1 Hochschild Homology ..................................... 8
1.2 The Trace Map and Morita Invariance ...................... 16
1.3 Derivations, Differential Forms ............................. 23
1.4 Nonunital Algebras and Excision ........................... 29
1.5 Hochschild Cohomology, Cotrace, Duality ................... 37
1.6 Simplicial Modules ....................................... 44
Bibliographical Comments on Chapter 1 .................... 48
Chapter 2. Cyclic Homology of Algebras .................... 51
2.1 Definition of Cyclic Homology ............................. 53
2.2 Connes' Exact Sequence, Morita Invariance, Excision ......... 61
2.3 Differential Forms, de Rham Cohomology ................... 68
2.4 Cyclic Cohomology ....................................... 72
2.5 Cyclic Modules .......................................... 75
2.6 Non-commutative Differential Forms ........................ 83
Bibliographical Comments on Chapter 2 ......... . . . . . . . . . . . 88
Chapter 3. Smooth Algebras and Other Examples .......... 89
3.1 Tensor Algebra .......................................... 90
3.2 Symmetric Algebras ...................................... 94
3.3 Universal Enveloping Algebras of Lie Algebras ............... 97
3.4 Smooth Algebras ......................................... 101
3.5 Andre-Quillen Homology .................................. 107
3.6 Deligne Cohomology ...................................... 112
Bibliographical Comments on Chapter 3 .................... 114
XII Contents
Chapter 4. Operations on Hochschild and Cyclic Homology 115
4.1 Conjugation and Derivation ............................... 116
4.2 Shuffle Product in Hochschild Homology .................... 123
4.3 Cyclic Shuffles and Kiinneth Sequence for He ............... 127
4.4 Product, Coproduct in Cyclic Homology .................... 133
4.5 ,X-Decomposition for Hochschild Homology .................. 139
4.6 ,X-Decomposition for Cyclic Homology ....................... 149
Bibliographical Comments on Chapter 4 .................... 155
Chapter 5. Variations on Cyclic Homology .................. 157
5.1 The Periodic and Negative Theories ........................ 158
5.2 Dihedral and Quaternionic Homology ....................... 167
5.3 Differential Graded Algebras ............................... 175
5.4 Commutative Differential Graded Algebras .................. 182
5.5 Bivariant Cyclic Cohomology .............................. 190
5.6 Topological Algebras, Entire Cyclic Cohomology ............. 195
Bibliographical Comments on Chapter 5 ........ . . . . . . . . . . .. 199
Chapter 6. The Cyclic Category, Tor and Ext Interpretation 201
6.1 Connes Cyclic Category ..de and the Category..dS ........... 202
6.2 Tor and Ext Interpretation of HH and He .................. 211
6.3 Crossed Simplicial Groups ................................. 214
6.4 The Category of Finite Sets and the ,X-Decomposition ......... 220
Bibliographical Comments on Chapter 6 .................... 225
Chapter 7. Cyclic Spaces and Sl-Equivariant Homology .... 227
7.1 Cyclic Sets and Cyclic Spaces .............................. 228
7.2 Cyclic Homology and Sl-Equivariant Homology .............. 234
7.3 Examples of Cyclic Sets and the Free Loop Space ............ 241
7.4 Hochschild Homology and Cyclic Homology of Group
Algebras ................................................ 250
Bibliographical Comments on Chapter 7 .................... 256
Chapter 8. Chern Character ................................ 257
8.1 The Classical Chern Character it la Chern-Weil .............. 258
8.2 The Grothendieck Group Ko ............................... 261
8.3 The Chern Character from Ko to Cyclic Homology ........... 264
8.4 The Dennis Trace Map and the Generalized Chern Character .. 269
8.5 The Bass Trace Conjecture and the Idempotent Conjecture .... 277
Bibliographical Comments on Chapter 8 .................... 279
Contents XIII
Chapter 9. Classical Invariant Theory ...................... 281
9.1 The Fundamental Theorems of Invariant Theory ............. 282
9.2 Coinvariant Theory and the Trace Map ..................... 284
9.3 Cayley-Hamilton and Amitsur-Levitzki Formulas ............. 287
9.4 Proofs of the Fundamental Theorems ....................... 291
9.5 Invariant Theory for the Orthogonal and Symplectic Groups ... 293
Bibliographical Comments on Chapter 9 .................... 297
Chapter 10. Homology of Lie Algebras of Matrices 299
10.1 Homology of Lie Algebras ................................. 301
10.2 Homology of the Lie Algebra gl(A) ......................... 306
10.3 Stability and First Obstruction to Stability .................. 315
10.4 Homology with Coefficients in the Adjoint Representation ..... 320
10.5 The Symplectic and Orthogonal Cases ...................... 323
10.6 Non-commutative Homology (or Leibniz Homology)
of the Lie Algebra of Matrices ............................. 326
Bibliographical Comments on Chapter 10 ....... . . . . . . . . . . .. 340
Chapter 11. Algebraic K-Theory ........................... 341
11.1 The Bass-Whitehead Group Kl and the Milnor Group K2 ..... 342
11.2 Higher Algebraic K -Theory ................................ 349
11.3 Algebraic K-Theory and Cyclic Homology of Nilpotent Ideals .. 361
11.4 Absolute and Relative Chern Characters .................... 371
11.5 Secondary Characteristic Classes ........................... 375
Bibliographical Comments on Chapter 11 ................... 379
Chapter 12. Non-commutative Differential Geometry ....... 381
12.1 Foliations and the Godbillon-Vey Invariant .................. 382
12.2 Fredholm Modules and Index Theorems ..................... 385
12.3 Novikov Conjecture on Higher Signatures .................... 390
12.4 The K -Theoretic Analogue of the Novikov Conjecture ........ 395
Chapter 13. Mac Lane (co)homology 399
by Jean-Louis Loday and Teimuraz Pirashvili
13.1 (Co)homology with Coefficients in Non-additive Bimodules .... 401
13.2 Mac Lane (co)homology ................................... 408
13.3 Stable K-theory and Mac Lane homology .................... 417
13.4 Calculations ............................................. 428
Bibliographical Comments on Chapter 13 ................... 442
References of Chapter 13 .................................. 443