Mixed Motives and Algebraic K-Theory
The relations that could or should exist between algebraic cycles, algebraic K-theory, and the cohomology of - possibly singular - varieties, are the topic of investigation of this book. The author proceeds in an axiomatic way, combining the concepts of t
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UweJannsen
Mixed Motives and Algebraic K-Theory
Springer-Verlag Berlin Heidelberg NewYork LondonParis Tokyo Hong Kong
Author
UweJannsen Max-Planck-Institut fur Mathematik Gottfried-Claren-Str. 26 5300 Bonn 3, Federal Republic of Germany
Mathematics Subject Classification (1980): Primary: 14A20, 14C30, 14G13, 18F25 Secondary: 12A67, 14C35, 14F15 ISBN 3-540-52260-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52260-3 Springer-Verlag New York Berlin Heidelberg
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 other ways. and storage in data banks. Duplication ofthis publication or parts thereof is only permitted under the provisions ofthe German Copyright Law of September 9. 1965, in its version of June 24. 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1990 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 Printed on acid-free paper
Preface
This is an almost unchanged version of my 1988 Habilitationsschrift at Regensburg. My original plan was to completely rewrite it for publication; in particular I wanted to make it more readable for the nonexpert. Finally I chose to rather publish it like it is than turn it into a long range project. So I have only made some minor corrections and added three appendices. The first one reproduces a letter from S. Bloch to me and the second one consists of an example by C. Schoen. I thank both for the permission to publish this material, and the latter for the effort of rewriting the example, which also figured in a letter to me. The third appendix contains some remarks and complements written in 1989.
Uwe Jannsen Bonn, November 1989
Introduction
This
text
consists of three parts. In part I we define a
category of mixed motives in the setting of absolute Hodge cycles. In part II we investigate, as general as possible, relations between algebraic cycles, algebraic K-theory, and mixed structures in the cohomology of arbitrary varieties. In part III we present some conjectures on Chern characters from K-theory into £-adic cohomology for varieties over finite fields or global fields, and prove these in some (very) specific cases.
Background
The concept of motives [Ma]
,[Kl]
, [SR] was introduced
by Grothendieck to explain phenomena in different cohomology theories of algebraic varieties in a coherent way, in particular those related to algebraic cycles and weights. For example in both the £-adic and the Hodge theory the cohomology Hi(X) of a smooth projective variety is pure of weight i, the class of an algebraic cycle of codimension j can be interpreted as a morphism from the 2'
trivial structure into H J(X) (j), and the parallel formulation of the conjectures of Hodge and of Tate is that the functor se
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