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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Readership

Graduate students and research mathematicians interested in algebraic geometry and algebraic topology.

Reviews

"The translation should be of help to motivic homotopy theorists who read English more easily than French."

-- Mathematical Reviews