Mixed Motives and Their Realization in Derived Categories
The conjectural theory of mixed motives would be a universal cohomology theory in arithmetic algebraic geometry. The monograph describes the approach to motives via their well-defined realizations. This includes a review of several known cohomology theori
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
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Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Tokyo
Annette Huber
Mixed Motives and their Realization in Derived Categories
Springer
Author Annette Huber Mathematisches Institut Universitat Munster Einsteinstr. 62 D-48 149 Munster, Germany E-mail: [email protected]
Mathematics Subject Classification (1991): 14A20, 14F99, 19F27, 18010
ISBN 3-540-59475-2 Springer-Verlag Berlin Heidelberg New York
CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera-ready output by the author SPIN: 10130310 46/3142-543210 - Printed on acid-free paper
Acknowledgement The reader will see from the mere number of citations how much this monograph owes to the ideas of Deligne, Beilinson and J annsen. The deep results on Hodge structures and the theory of weights in the ladic situation are due to Deligne. Without them my construction would break down. From his paper [12] I also take the techniques in working with simplicial varieties. The idea how to give a good definition of absolute cohomology without a formalism of sheaves is Beilinson's in his construction of absolute Hodge cohomology [3]. On the other hand my definition of Chern classes is built from a sketch in [2]. I got the necessary background on Ktheory from Thomason's papers. I am particularly thankful for his patient explanations during a visit in Munster. Proposition 18.1.5 is due to him. The systematic treatment of mixed realizations was introduced in the book [39] by Jannsen. My work starts from the considerations in his §6. I thank him for his encouraging interest in my work and for a number of useful comments. I thank Prof. Deligne for pointing out to me a number of misconceptions in an earlier version [18]. I very much appreciate the help of Prof. Scholl who corrected my english grammar. I thank him heartily. Finally, I am very much indebted to Prof. Deninger. He acquainted me with the theory of motives and their realizations. His interest was the starting point for the project of my research. In particular the sections on Iextensions would not have been written without his questions. I thank him for constant encouragement and for making the working atmosphere in Munster what it is. G. Kings and M. Schroter were always ready to discuss problems I had. K. Kiinnemann took on the tedious task
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