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Algebraic Cobordism (Springer Monographs in Mathematics)by Marc Levine
Synopses & ReviewsPublisher 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 ContentsIntroduction. 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 BorelMoore 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 Ktheory. 4.3. The cobordism ring of a point. 4.4. Degree formulas. 4.5. Comparison with the Chow groups. V. Oriented BorelMoore homology. 5.1. Oriented BorelMoore homology theories. 5.2. Other oriented theories. VI. Functoriality. 6.1. Refined cobordism. 6.2. Intersection with a pseudodivisor. 6.3. Intersection with a pseudodivisor II. 6.4. A moving lemma. 6.5. Pullback for l.c.i. morphisms. 6.6. Refined pullback and refined intersections. VII. The universality of algebraic cobordism. 7.1. Statement of results. 7.2. Pullback in BorelMoore homology theories. 7.3. Universality 7.4. Some applications. Appendix A: Resolution of singularities. References. Index. Glossary of Notation.
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