A^1-Homotopy Theory

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Rouintan, Ahmad | 2022

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 55132 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Rastegar, Arash
Abstract:
A^1-homotopy theory or the motivic homotopy theory is a homotopy theory for smooth schemes of finite type over a Noetherian base scheme of finite dimension. A^1-homotopy was introduced by Vladimir Voevodsky and Fabien Morel in the 90s. The fundamental idea of this theory is that for schemes the affine line A^1 should play the role of the interval I = [0,1]. A^1-homotopy has been proved to be an important theory. Some familiar cohomology theories like motivic cohomology and algebraic K-theory are representable in this theory. Motivic cohomology and A^1-homotopy theory also appear in Voevodsky’s proof of Milnor’s conjecture and Block-Kato conjecture. In this thesis,we will give an introduction to the A^1-homotopy theory with a focus on the analogy between this theory and the classical homotopy theory. Then, we will discuss some important recent works in this theory
Keywords:
Homotopy Theory ; Motivic Cohomology ; Motivic Homotopy Theory ; Algebraic K-Theory