Fabien Morel Poradie kníh

This text deals with A 1 -homotopy theory over a base field, i. e., with the natural homotopy theory associated to the category of smooth varieties over a field in which the affine line is imposed to be contractible. It is a natural sequel to the foundational paper on A 1 -homotopy theory written together with V. Voevodsky. Inspired by classical results in algebraic topology, we present new techniques, new results and applications related to the properties and computations of A 1 -homotopy sheaves, A 1 -homology sheaves, and sheaves with generalized transfers, as well as to algebraic vector bundles over affine smooth varieties.

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.