Algebraic Cobordism

By: Marc Levine, Fabien Morel

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Hardcover | 9 January 2007

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260 Pages

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24.13 x 16.51 x 1.91

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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.

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Cobordism and oriented cohomology	p. 1
Oriented cohomology theories	p. 1
Algebraic cobordism	p. 6
Relations with complex cobordism	p. 13
The definition of algebraic cobordism	p. 17
Oriented Borel-Moore functors	p. 17
Oriented functors of geometric type	p. 26
Some elementary properties	p. 29
The construction of algebraic cobordism	p. 35
Some computations in algebraic cobordism	p. 40
Fundamental properties of algebraic cobordism	p. 51
Divisor classes	p. 51
Localization	p. 55
Transversality	p. 66
Homotopy invariance	p. 75
The projective bundle formula	p. 77
The extended homotopy property	p. 80
Algebraic cobordism and the Lazard ring	p. 83
Weak homology and Chern classes	p. 83
Algebraic cobordism and K-theory	p. 105
The cobordism ring of a point	p. 116
Degree formulas	p. 122
Comparison with the Chow groups	p. 136
Oriented Borel-Moore homology	p. 141
Oriented Borel-Moore homology theories	p. 141
Other oriented theories	p. 149
Functoriality	p. 159
Refined cobordism	p. 159
Intersection with a pseudo-divisor	p. 169
Intersection with a pseudo-divisor II	p. 178
A moving lemma	p. 184
Pull-back for l.c.i. morphisms	p. 192
Refined pull-back and refined intersections	p. 201
The universality of algebraic cobordism	p. 215
Statement of results	p. 215
Pull-back in Borel-Moore homology theories	p. 217
Universality	p. 221
Some applications	p. 225
A Resolution of singularities	p. 233
References	p. 237
Index	p. 239
Glossary of Notation	p. 243
Table of Contents provided by Publisher. All Rights Reserved.

Algebraic Cobordism

At a Glance

Hardcover

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How to return your order

Defective items

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