Levine / Morel Algebraic Cobordism

1. Auflage 2007
ISBN: 978-3-540-36824-3
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 246 Seiten

Reihe: Springer Monographs in Mathematics

Following Quillen's approach to complex cobordism, the authors introduce the notion of oriented cohomology theory on the category of smooth varieties over a fixed field. They prove the existence of a universal such theory (in characteristic 0) called Algebraic Cobordism. Surprisingly, this theory satisfies the analogues of Quillen's theorems: the cobordism of the base field is the Lazard ring and the cobordism of a smooth variety is generated over the Lazard ring by the elements of positive degrees. This implies in particular the generalized degree formula conjectured by Rost. The book also contains some examples of computations and applications.

Autoren/Hrsg.

Levine, Marc

Morel, Fabien

Inhaltsverzeichnis

1;Introduction;6
2;Contents;10
3;1 Cobordism and oriented cohomology;12
3.1;1.1 Oriented cohomology theories;12
3.2;1.2 Algebraic cobordism;17
3.3;1.3 Relations with complex cobordism;24
4;2 The definition of algebraic cobordism;27
4.1;2.1 Oriented Borel-Moore functors;27
4.2;2.2 Oriented functors of geometric type;36
4.3;2.3 Some elementary properties;39
4.4;2.4 The construction of algebraic cobordism;45
4.5;2.5 Some computations in algebraic cobordism;50
5;3 Fundamental properties of algebraic cobordism;60
5.1;3.1 Divisor classes;60
5.2;3.2 Localization;64
5.3;3.3 Transversality;75
5.4;3.4 Homotopy invariance;84
5.5;3.5 The projective bundle formula;86
5.6;3.6 The extended homotopy property;89
6;4 Algebraic cobordism and the Lazard ring;92
6.1;4.1 Weak homology and Chern classes;92
6.2;4.2 Algebraic cobordism and K-theory;114
6.3;4.3 The cobordism ring of a point;125
6.4;4.4 Degree formulas;131
6.5;4.5 Comparison with the Chow groups;145
7;5 Oriented Borel-Moore homology;150
7.1;5.1 Oriented Borel-Moore homology theories;150
7.2;5.2 Other oriented theories;158
8;6 Functoriality;168
8.1;6.1 Re.ned cobordism;168
8.2;6.2 Intersection with a pseudo-divisor;178
8.3;6.3 Intersection with a pseudo-divisor II;187
8.4;6.4 A moving lemma;193
8.5;6.5 Pull-back for l.c.i. morphisms;201
8.6;6.6 Refined pull-back and re.ned intersections;210
9;7 The universality of algebraic cobordism;223
9.1;7.1 Statement of results;223
9.2;7.2 Pull-back in Borel-Moore homology theories;225
9.3;7.3 Universality;229
9.4;7.4 Some applications;233
10;A Resolution of singularities;240
11;References;243
12;Index;245
13;Glossary of Notation;249