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Algebraic Cobordism (Springer Monographs in Mathematics)

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Synopses & Reviews

Publisher Comments:

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.

Table of Contents

Introduction.- I. Cobordism and oriented cohomology.- 1.1. Oriented cohomology theories. 1.2. Algebraic cobordism. 1.3. Relations with complex cobordism. - II. The definition of algebraic cobordism. 2.1. Oriented Borel-Moore functions. 2.2. Oriented functors of geometric type. 2.3. Some elementary properties. 2.4. The construction of algebraic cobordism. 2.5. Some computations in algebraic cobordism.- III. Fundamental properties of algebraic cobordism. 3.1. Divisor classes. 3.2. Localization. 3.3. Transversality. 3.4. Homotopy invariance. 3.5. The projective bundle formula. 3.6. The extended homotopy property. IV. Algebraic cobordism and the Lazard ring. 4.1. Weak homology and Chern classes. 4.2. Algebraic cobordism and K-theory. 4.3. The cobordism ring of a point. 4.4. Degree formulas. 4.5. Comparison with the Chow groups. V. Oriented Borel-Moore homology. 5.1. Oriented Borel-Moore homology theories. 5.2. Other oriented theories.- VI. Functoriality. 6.1. Refined cobordism. 6.2. Intersection with a pseudo-divisor. 6.3. Intersection with a pseudo-divisor II. 6.4. A moving lemma. 6.5. Pull-back for l.c.i. morphisms. 6.6. Refined pull-back and refined intersections. VII. The universality of algebraic cobordism. 7.1. Statement of results. 7.2. Pull-back in Borel-Moore homology theories. 7.3. Universality 7.4. Some applications.- Appendix A: Resolution of singularities.- References.- Index.- Glossary of Notation.

Product Details

ISBN:
9783540368229
Author:
Levine, Marc
Publisher:
Springer
Author:
Morel, Fabien
Author:
Morel, F.
Author:
Levine, M.
Subject:
Geometry - Algebraic
Subject:
Algebra - General
Subject:
Chow ring
Subject:
Cobordism
Subject:
K-theory
Subject:
Oriented cohomology theory
Subject:
Algebraic Geometry
Subject:
Commutative Rings and Algebras
Subject:
Mathematics-Algebraic Geometry
Copyright:
Edition Description:
Book
Series:
Springer Monographs in Mathematics
Publication Date:
20070221
Binding:
HARDCOVER
Language:
English
Pages:
258
Dimensions:
235 x 155 mm

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Related Subjects


Science and Mathematics » Mathematics » Algebra » General
Science and Mathematics » Mathematics » Combinatorics
Science and Mathematics » Mathematics » General
Science and Mathematics » Mathematics » Geometry » Algebraic Geometry
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