Homotopy Theory of Schemes

In this text, the author presents a general framework for applying the standard methods from homotopy theory to the category of smooth schemes over a reasonable base scheme $k$. He defines the homotopy category $h(mathcal{E}_k)$ of smooth $k$-schemes and shows that it plays the same role for smooth $k$-schemes as the classical homotopy category plays for differentiable varieties. It is shown that certain expected properties are satisfied, for example, concerning the algebraic $K$-theory of those schemes. In this way, advanced methods of algebraic topology become available in modern algebraic geometry.