Arbeitsgemeinschaft mit aktuellem Thema: Algebraic Cobordism

Abstract

The aim of this Arbeitsgemeinschaft was to present the theory of Algebraic Cobordism due to Marc Levine and Fabien Morel through the lines of their original articles: Inspired by the work of Quillen on complex cobordism, one first introduces the notion of oriented cohomology theory on the category of smooth varieties over a field k. Grothendieck’s method allows one to extend the theory of Chern classes to such theories. When char(k) = 0, one proves the existence of a universal oriented cohomology theory X → Ω∗ (X). Localisation and homotopy invariance are then proved for this universal theory. For any field k of characteristic 0 one can prove for algebraic cobordism the analogue of a theorem of Quillen on complex cobordism: the cobordism ring of the ground field is the Lazard ring L and for any smooth k-variety X, the algebraic cobordism ring Ω∗ (X) is generated, as an L-module, by elements of non negative degree. This implies Rost’s conjectured degree formula. One also gives a relation between the Chow ring, the K0 of a smooth k-variety X and Ω∗ (X). The technical construction of pullbacks is the subject of two talks. At the end one presents the state of advances on the conjectural isomorphism between Levine-Morel construction of algebraic cobordism and the ”homotopical algebraic cobordism”, the cohomology theory represented by motivic Thom spectrum in the Morel-Voevodsky A1 -stable homotopy category. The Arbeitsgemeinschaft was organised by Marc Levine(Boston) and Fabien Morel(München). It was well attended with over 40 participants.