Algebraic Cobordism

By M. Levine. 2003

295 citations recorded by the source corpus.

Together with F. Morel, we have constructed in [6, 7, 8] a theory of algebraic cobordism, an algebro-geometric version of the topological theory of complex cobordism. In this paper, we give a survey of the construction and main results of this theory; in the final section, we propose a candidate for a theory of higher algebraic cobordism, which hopefully agrees with the cohomology theory represented by the P-spectrum MGL in the Morel-Voevodsky stable homotopy category. 2000 Mathematics Subject Classification: 19E15, 14C99, 14C25.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/788a53fa055aaaa41abb6f79230b6472178ff43f

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Citations in the corpus

Motivic Landweber exactness (2008, methodology): “the works of Hopkins-Morel reviewed in [19] and Voevodsky [41].”
Schubert calculus for algebraic cobordism (2009, methodology): “We briefly recall the geometric definition of algebraic cobordism [15] and some of its basic properties as established in [14].”
Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics (2019, methodology): “The calculations that we have presented, which may be viewed as the beginning of the study of the virtual Euler characteristics of Quot schemes of surfaces, already show some features of Vafa-Witten theory: the appearance of the Kawai-Yoshioka formula (in 8See [27] for a foundational treatment of al”
Equivariant oriented cohomology of flag varieties (2014, methodology): “2] (except that h takes values in rings, not in graded rings) with push-forwards and characteristic classes being as in [LM07].”
Invariants, torsion indices and oriented cohomology of complete flags (2009, background): “A G ] 8 A pr 2 01 0 INVARIANTS, TORSION INDICES AND ORIENTED COHOMOLOGY OF COMPLETE FLAGS”
On cohomological Hall algebras of quivers: Generators (2017, background): “The functors MG ∗ , M ∗ G satisfy the same properties as M∗, M ∗, for which we refer to [5], [13], [15], [23].”
Around the Gysin triangle. II. (2007, background): “The induction relies on the following lemma due to Morel.”
Algebraic cobordism revisited (2006, background): “The goal is to define a universal oriented Borel-Moore cohomology theory of schemes.”
The special fiber of the motivic deformation of the stable homotopy category is algebraic (2018, background): “The motivic spectrum MGL is the algebraic cobordism spectrum introduced by Voevodsky [69] and studied by Levine–Morel [39], Panin–Pimenov–Röndigs [58] and many others.”