In mathematics, algebraic cobordism is an analogue of complex cobordism for smooth quasi-projective schemes over a field. It was introduced by Marc Levine and Fabien Morel (2001, 2001b).
An oriented cohomology theory on the category of smooth quasi-projective schemes Sm over a field k consists of a contravariant functor A* from Sm to commutative graded rings, together with push-forward maps f* whenever f:Y→X has relative dimension d for some d. These maps have to satisfy various conditions similar to those satisfied by complex cobordism. In particular they are "oriented", which means roughly that they behave well on vector bundles; this is closely related to the condition that a generalized cohomology theory has a complex orientation.
Over a field of characteristic 0, algebraic cobordism is the universal oriented cohomology theory for smooth varieties. In other words there is a unique morphism of oriented cohomology theories from algebraic cobordism to any other oriented cohomology theory.
Levine (2002) and Levine & Morel (2007) give surveys of algebraic cobordism.
The algebraic cobordism ring of generalized flag varieties has been computed by Hornbostel & Kiritchenko (2011).
References
edit- Hornbostel, Jens; Kiritchenko, Valentina (2011), "Schubert calculus for algebraic cobordism", J. Reine Angew. Math., 656: 59–85, arXiv:0903.3936, doi:10.1515/CRELLE.2011.043, MR 2818856
- Levine, M (2002), "Algebraic cobordism", in Li, Tatsien (ed.), Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Beijing: Higher Ed. Press, pp. 57–66, ISBN 978-7-04-008690-4, MR 1957020, archived from the original on 2011-08-20, retrieved 2011-06-30
- Levine, Marc; Morel, Fabien (2001), "Cobordisme algébrique. I", Comptes Rendus de l'Académie des Sciences, Série I, 332 (8): 723–728, Bibcode:2001CRASM.332..723L, doi:10.1016/S0764-4442(01)01832-8, ISSN 0764-4442, MR 1843195
- Levine, Marc; Morel, Fabien (2001), "Cobordisme algébrique. II", Comptes Rendus de l'Académie des Sciences, Série I, 332 (9): 815–820, Bibcode:2001CRASM.332..815L, doi:10.1016/S0764-4442(01)01833-X, ISSN 0764-4442, MR 1836092
- Levine, M; Morel, Fabien (2007), Algebraic cobordism, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-36824-8, ISBN 978-3-540-36822-9, MR 2286826