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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.
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Table of Contents

Introduction The homotopic category Homotopic excision, homotopic purity and projective blow-ups Homotopic classification of vector bundles Appendix A: Review of homotopic algebra Appendix B: Ample families of invertible bundles on a scheme References.

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