Homotopy Theory of Schemes
The aim of this memoir is to define the homotopy category of (Noetherian, separated and admitting an ample family) schemes and to show that this category plays the same role for smooth (Noetherian, separated and admitting an ample family) schemes as the classical homotopy category does for differentiable (topological) varieties. For schemes of finite Krull dimension, the combinatorial definition of Morel agrees with the topological approach (based on the concept of sheaves in the Nisnevich topology) of Voevodsky. The main results of this monograph are the homotopic purity theorems, which are the foundation of localization exact sequences for any oriented cohomology theory and Poincare duality; the Thom space of closed immersions between two smooth schemes of the above type depends only on the normal bundle of the immersion. The main technical obstacle in comparison to the case of differentiable varieties is that one does not have the notion of a tubular neighbourhood. A suitable replacement of this device relies on techniques of deformations of the normal bundle.