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Algebraic Cobordism
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 theo...
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| Auteurs principaux : | , |
|---|---|
| Format : | Livre |
| Langue : | anglais |
| Titre complet : | Algebraic Cobordism / Marc Levine, Fabien Morel. |
| Édition : | 1st ed. 2007. |
| Publié : |
Berlin, Heidelberg :
Springer Berlin Heidelberg
, [20..] Cham : Springer Nature |
| Collection : | Springer monographs in mathematics (Internet) |
| Accès en ligne : |
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| Condition d'utilisation et de reproduction : | Conditions particulières de réutilisation pour les bénéficiaires des licences nationales : https://www.licencesnationales.fr/springer-nature-ebooks-contrat-licence-ln-2017 |
| Reproduction de : | Numérisation de l'édition de Berlin ; Heidelberg ; New York : Springer , cop.2007 |
| Contenu : | 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 |
| Sujets : | |
| Documents associés : | Autre format:
Algebraic cobordism Autre format: Algebraic Cobordism Autre format: Algebraic Cobordism Autre format: Algebraic cobordism |
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| 200 | 1 | |a Algebraic Cobordism |f Marc Levine, Fabien Morel. | |
| 205 | |a 1st ed. 2007. | ||
| 214 | 0 | |a Berlin, Heidelberg |c Springer Berlin Heidelberg | |
| 214 | 2 | |a Cham |c Springer Nature |d [20..] | |
| 225 | 0 | |a Springer Monographs in Mathematics |x 2196-9922 | |
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| 324 | |a Numérisation de l'édition de Berlin ; Heidelberg ; New York : Springer , cop.2007 | ||
| 327 | 1 | |a Introduction |a I. Cobordism and oriented cohomology |a 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 |a 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 |a 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 |a Appendix A: Resolution of singularities |a References |a Index |a Glossary of Notation | |
| 330 | |a 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 | ||
| 371 | 0 | |a Accès en ligne pour les établissements français bénéficiaires des licences nationales | |
| 371 | 0 | |a Accès soumis à abonnement pour tout autre établissement | |
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| 452 | | | |t Algebraic Cobordism |b Texte imprimé |y 9783642071911 | |
| 452 | | | |t Algebraic Cobordism |b Texte imprimé |y 9783540827214 | |
| 452 | | | |0 113721137 |t Algebraic cobordism |f M. Levine, F. Morel |c Berlin |n Springer |d 2007 |p 1 vol. (XII-244 p.) |s Springer monographs in mathematics |y 978-3-540-36822-9 | |
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