Homology of Linear Groups
Daniel Quillen's definition of the higher algebraic K-groups of a ring emphasized the importance of computing the homology of groups of matrices. This text traces the development of this theory from Quillen's fundamental calculation of the cohomology of G
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Series Editors H. Bass J. Oesterle A. Weinstein
Kevin P. Knudson Homology of Linear Groups
Springer Basel AG
Author: Kevin P. Knudson Department of Mathematics Wayne State University Detroit, MI 48202 USA e-mail: [email protected] 2000 Mathematics Subject Classification 20G I0
Knudson, Kevin P. (Patrick), 1969Homology of linear groups / Kevin P. Knudson. p. cm. -- (Progress in mathematics ; v. 193) Includes bibliographical references and index. ISBN 978-3-0348-9523-1 ISBN 978-3-0348-8338-2 (eBook) DOI 10.1007/978-3-0348-8338-2 I. Linear algebraic groups. 2. Homology theory. I. Title. 11. Progress in mathematics (Boston, Mass.) ; vol. 193. QAI79 .K59 2000 512'.55--dc21
00-057147
Deutsche Bibliothek Cataloging-in-Publication Data Knudson, Kevin P.: Homology oflinear groups / Kevin P. Knudson. - Basel; Boston; Berlin : Birkhäuser, 2001 (Progress in mathematics ; Vol. 193) ISBN 978-3-0348-9523-1
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2001 Springer Basel AG Originally published by Birkhäuser Verlag in 2001 Softcover reprint ofthe hardcover 1st edition 2001 Printed on acid-free paper produced of chlorine-free pulp. TCF co ISBN 978-3-0348-9523-1 987654321
To Ellen, for inspiration; and to Gus, for distraction
Contents Preface ................................................................
IX
Chapter 1. Topological Methods ...................................... 1.1. Finite Fields ................... . . . ..... .. ...... .. ...... .. ..... 1.2. Quillen's Conjecture .......................................... 1.3. Etale homotopy theory... . .. .... . ..... . . ........ ......... .. . .. 1.4. Analytical Methods ......................... . ................. 1.5. Unstable Calculations ..... ..... . ............... . ...... .. ...... 1.6. Congruence Subgroups..... ........ ....... . . ..... ........... . . Exercises
1 1 12 14 19 21 23 29
Chapter 2. Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. van der Kallen's Theorem..................................... 2.2. Stability for rings with many units ...................... . ..... 2.3. Local rings and Milnor K -theory .............................. 2.4. Auxiliary stability results ..................................... 2.5. Stability via Homotopy .................................. .. .... 2.6. The Rank Conjecture ........ . . .. ............................. Exercises ...........................................................
33 34 38 46 56 59 61 63
Chapter 3. Low-dimensional Results .................................. 3.1. Scissors Congruence ..... . ..................................... 3.2. The Bloch Group ... .... .. ................ .. .. . .... .. ..... .. .. 3.3. Ext
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