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Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes and Exotic Motivic Spheres

Krishna Kumar Madhavan Vijayalakshmi

Abstract

One of the emerging problems in algebraic geometry is to characterize the affine $n$-space $\mathbb{A}^n$ among smooth affine schemes up to $\mathbb{A}^1$-contractibility. Recent efforts show that this characterization holds in dimensions $n<3$ over certain fields. In this thesis, we extend this observation to "reasonably" arbitrary base schemes in relative dimensions $d<3$, exploiting the Zariski local triviality and the triviality of the sheaf of relative differentials. From dimensions $n\geq 3$, the existence of smooth "exotic" affine schemes - those that are $\mathbb{A}^1$-contractible but not isomorphic to the affine $n$-space - has already been established. A well-studied family constitutes the Koras-Russell threefolds $\mathcal{K}$ and their higher-dimensional prototypes $\mathcal{X}_n$, whose $\mathbb{A}^1$-contractibility has been so far proven over fields of characteristic zero. Here, we extend the relative $\mathbb{A}^1$-contractibility of $\mathcal{K}$ and $\mathcal{X}_n$ over a Noetherian base scheme in arbitrary dimensions. Then, using these prototypes, we study the existence of "exotic spheres" - $n$-dimensional smooth schemes that are $\mathbb{A}^1$-homotopic, but not isomorphic to $\mathbb{A}^n \backslash \{0\}$ - in motivic homotopy theory. This result can be seen as the "compact" analog of the study of exotic affine schemes. Our main result shows that in all dimensions $n\geq 4$, the quasi-affine varieties $\mathcal{X}_n \backslash \{\bullet\}$ give a model for the exotic motivic spheres over infinite perfect fields. The novelty is that these constitute the first family of examples of smooth motivic spheres of dimension $n$, which are not isomorphic to $\mathbb{A}^n \backslash \{0\}$.

Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes and Exotic Motivic Spheres

Abstract

One of the emerging problems in algebraic geometry is to characterize the affine -space among smooth affine schemes up to -contractibility. Recent efforts show that this characterization holds in dimensions over certain fields. In this thesis, we extend this observation to "reasonably" arbitrary base schemes in relative dimensions , exploiting the Zariski local triviality and the triviality of the sheaf of relative differentials. From dimensions , the existence of smooth "exotic" affine schemes - those that are -contractible but not isomorphic to the affine -space - has already been established. A well-studied family constitutes the Koras-Russell threefolds and their higher-dimensional prototypes , whose -contractibility has been so far proven over fields of characteristic zero. Here, we extend the relative -contractibility of and over a Noetherian base scheme in arbitrary dimensions. Then, using these prototypes, we study the existence of "exotic spheres" - -dimensional smooth schemes that are -homotopic, but not isomorphic to - in motivic homotopy theory. This result can be seen as the "compact" analog of the study of exotic affine schemes. Our main result shows that in all dimensions , the quasi-affine varieties give a model for the exotic motivic spheres over infinite perfect fields. The novelty is that these constitute the first family of examples of smooth motivic spheres of dimension , which are not isomorphic to .
Paper Structure (153 sections, 131 theorems, 308 equations, 2 figures)

This paper contains 153 sections, 131 theorems, 308 equations, 2 figures.

Table of Contents

  1. Abstract
  2. Preface
  3. Affine Algebraic Geometry
  4. The Flashback
  5. Some Phenomenal Applications of Motivic Homotopy Theory
  6. Acknowledgments
  7. Useful notations
  8. Overview
  9. Introduction
  10. Topology at Infinity
  11. Homology at infinity
  12. Homotopy at infinity
  13. Smooth Affine Schemes up to $\mathbb{A}^1$-Contractibility
  14. Exotic Affine Varieties
  15. Exotic complex surfaces
  16. ...and 138 more sections

Key Result

Theorem 1

Let $X$ be an étale scheme of finite presentation over an arbitrary base scheme $S$. Then the canonical morphism $f: X\to S$ is a relative $\mathbb{A}^1$-weak equivalence in $\operatorname{Spc}_S$ if and only if $f$ is an isomorphism in $\mathrm{Sm}_S$.

Figures (2)

  • Figure 1: Whitehead Continuum, Tsai-Whitehead
  • Figure 2: Fibers of $\mathcal{K}$ when projected to $x$-axis; the generic fiber is shown in red

Theorems & Definitions (363)

  • Theorem
  • Theorem
  • Corollary
  • Corollary
  • Theorem
  • Corollary
  • Corollary
  • Theorem
  • Proposition
  • Theorem
  • ...and 353 more