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algebraic cobordism

Contents

Idea

Oriented Borel-Moore homology is a generalization of the theory CH *:Sch KAbGrp *CH_* : Sch_K \to AbGrp_* of Chow groups on schemes over a field KK. The additivity of the first Chern class operator (the formula c 1(LM)=c 1(L)+c 1(M)c_1(L \otimes M) = c_1(L) + c_1(M)), is replaced by imposing a formal group law FF, i.e. c 1(LM)=F(c 1(L),c 1(M))c_1(L \otimes M) = F(c_1(L), c_1(M)). Algebraic cobordism is the universal oriented Borel-Moore homology theory. This is a direct analogue of the complex cobordism MU *MU^* of Quillen.

(Co)homological theories for algebraic varieties have led Grothendieck to propose a theory of motives which has partly been realized (e.g. in its triangulated and dg-versions) so far. One systematic approach is to study the homotopy theory in this context, namely 𝔸 1\mathbb{A}^1-homotopy theory. However in practice, one is led to calculate invariants and it is difficult to find new computable cases; the study of algebraic cycles in higher codimension is notoriously difficult, and the Chow groups are difficult to compute. Somewhat more accessible is algebraic K-theory. This is unlike the classical situation where in addition many other extraordinary (co)homology theories are at hand. In the abstract setup, Voevodsky and others did study more general extraordinary cohomology theories, but down-to-Earth examples are difficult to find. This was roughly the motivation for two people – Marc Levine and Fabien Morel – to construct in a single effort the notion of algebraic cobordism, which is an analogue of complex cobordism for algebraic varieties. Many of the tools from Quillen’s and Novikov’s work on complex cobordism, like Lazard’s ring and formal groups, play a major role.

Algebraic cobordism is the universal oriented Borel–Moore homology theory, where ‘oriented’ stands for the existence of the proper? direct image homomorphisms. The fact that 𝔸 1\mathbb{A}^1-homotopy theory is not used —that is, the constructions are more direct— is an advantage.

On the other hand a universal oriented motivic cohomology theory is not the algebraic cobordism of Morel and Levine but rather the cohomology associated to the motivic spectrum MGLMGL of Voevodsky, which is different and also may be called a theory of algebaric cobordism. An early different approach to some sort of algebraic cobordism is from late 1970-s work of Victor Snaith, discussed in that entry.

References

Introductory

The original announcements:

  • M. Levine, F. Morel, Cobordisme algébrique I, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 723–728, 2001 (doi01833-X)); Cobordisme algébrique II, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 815–820, 2001 (doi01832-8)).

A very complete book:

  • M. Levine, F. Morel, Algebraic cobordism, Springer 2007.

Short surveys:

Lecture notes:

An Oberwolfach workshop:

  • M. Levine, F. Morel, Oberwolfach Arbeitsgemeinschaft mit aktuellem Thema, April 2005 report, notes

Further work

An equivalent and simpler construction was given in

  • M. Levine, R. Pandharipande, Algebraic cobordism revisited, math.AG/0605196

The construction was extended to derived schemes in the paper

The close connection of algebraic cobordism with K-theory is discussed in

  • José Luis González, Kalle Karu. Universality of K-theory. 2013. arXiv:1301.3815.

An algebraic analogue of h-cobordism:

  • Aravind Asok, Fabien Morel, Smooth varieties up to 𝔸 1\mathbb{A}^1-homotopy and algebraic h-cobordisms, arXiv:0810.0324

Algebraic cobordism in mixed characteristic:

Revised on April 10, 2014 04:23:20 by Adeel Khan (132.252.63.38)