Spencer Bloch's 1979 Duke lectures, a milestone in modern mathematics, have been out of print almost since their first publication in 1980, yet they have remained influential and are still the best place to learn the guiding philosophy of algebraic cycles and motives. This edition, now professionally typeset, has a new preface by the author giving his perspective on developments in the field over the past 30 years. The theory of algebraic cycles encompasses such central problems in mathematics as the Hodge conjecture and the Bloch-Kato conjecture on special values of zeta functions. The book begins with Mumford's example showing that the Chow group of zero-cycles on an algebraic variety can be infinite-dimensional, and explains how Hodge theory and algebraic K-theory give new insights into this and other phenomena.
Author(s): Spencer Bloch
Edition: 2
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 156
Tags: Математика;Общая алгебра;Теория колец;
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
30 Years later.........Page 9
Motives......Page 10
Lecture 1: Zero-cycles......Page 14
Lectures 2 and 3: Intermediate jacobians......Page 16
Lecture 4: Cohomological methods......Page 17
Lecture 5: The conjecture of Milnor–Bloch–Kato......Page 18
Lecture 6: Infinitesimal methods in motivic cohomology......Page 19
Lectures 8 and 9: Regulators and values of L-functions......Page 20
Coda: Motives in physics......Page 22
References for preface......Page 23
LECTURES ON ALGEBRAIC CYCLES......Page 27
0 Introduction......Page 29
References for Lecture 0......Page 34
1 Zero-cycles on surfaces......Page 35
References for Lecture 1......Page 44
Appendix: On an argument of Mumford in the theory of algebraic cycles......Page 47
References for Lecture 1 Appendix......Page 50
2 Curves on threefolds and intermediate jacobians......Page 51
References for Lecture 2......Page 61
3 Curves on threefolds and intermediate jacobians – the relative case......Page 63
References for Lecture 3......Page 70
4 K-theoretic and cohomological methods......Page 71
References for Lecture 4......Page 82
5 Torsion in the Chow group......Page 85
References for Lecture 5......Page 93
6 Complements on H2 (K2)......Page 95
References for Lecture 6......Page 103
7 Diophantine questions......Page 105
References for Lecture 7......Page 121
8 Relative cycles and zeta functions......Page 123
References for Lecture 8......Page 140
9 Relative cycles and zeta functions – continued......Page 141
References for Lecture 9......Page 146
Bibliography......Page 149
Index......Page 155
