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.

Introduction....Pages 1-13

Unramified Sheaves and Strongly $${\mathbb{A}}^{1}$$ -Invariant Sheaves....Pages 15-48

Unramified MilnorWitt K-Theories....Pages 49-80

Geometric Versus Canonical Transfers....Pages 81-112

The RostSchmid Complex of a Strongly $${\mathbb{A}}^{1}$$ -Invariant Sheaf....Pages 113-148

$${\mathbb{A}}^{1}$$ -Homotopy Sheaves and $${\mathbb{A}}^{1}$$ -Homology Sheaves....Pages 149-175

$${\mathbb{A}}^{1}$$ -Coverings, $${\pi }_{1}^{{\mathbb{A}}^{1} }({\mathbb{P}}^{n})$$ and $${\pi }_{1}^{{\mathbb{A}}^{1} }(S{L}_{n})$$ ....Pages 177-197

$${\mathbb{A}}^{1}$$ -Homotopy and Algebraic Vector Bundles....Pages 199-207

The Affine B.G. Property for the Linear Groups and the Grassmanian....Pages 209-226

Back Matter....Pages 227-259

- Author: Fabien Morel (auth.)
- Edition: 1
- Publication Date: 2012
- Publisher: Springer-Verlag Berlin Heidelberg
- ISBN-10: 3642295134
- ISBN-13: 9783642295133
- Pages: 270
- Format: pdf
- Size: 3.3M