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Weil restriction, normal bundles and motivic Thom spaces

Xi Chen, Guangzhao Zhu

Abstract

Recent developments in motivic homotopy theory, especially the construction of norm functors by Bachmann and Hoyois, rely heavily on the machinery of infinite categories. In this paper, we take a purely geometric and elementary approach via the Weil restriction of schemes -- the fundamental geometric operation underlying these norm functors -- without invoking highly abstract categorical methods. We show that the Weil restriction preserves vector bundles and extend an existing result on normal bundles. We then construct the Weil restriction functor on the unstable motivic homotopy category and prove its compatibility with Thom spaces. Finally, in the setting of effective motives and the associated cohomology theories, we show that the Weil restriction sends Thom classes to Thom classes.

Weil restriction, normal bundles and motivic Thom spaces

Abstract

Recent developments in motivic homotopy theory, especially the construction of norm functors by Bachmann and Hoyois, rely heavily on the machinery of infinite categories. In this paper, we take a purely geometric and elementary approach via the Weil restriction of schemes -- the fundamental geometric operation underlying these norm functors -- without invoking highly abstract categorical methods. We show that the Weil restriction preserves vector bundles and extend an existing result on normal bundles. We then construct the Weil restriction functor on the unstable motivic homotopy category and prove its compatibility with Thom spaces. Finally, in the setting of effective motives and the associated cohomology theories, we show that the Weil restriction sends Thom classes to Thom classes.
Paper Structure (13 sections, 10 theorems, 71 equations)

This paper contains 13 sections, 10 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Norm functors and Weil restriction
  3. Basic properties of Weil restriction
  4. Relation to Norm functors
  5. Weil restriction, vector bundles and normal bundles
  6. Weil restriction of vector bundles
  7. Weil restriction of normal bundles
  8. Thom spaces and motivic category
  9. Weil restriction and unstable motivic homotopy category
  10. Weil restriction of Thom spaces
  11. Thom classes for effective motives
  12. Effective motives and oriented cohomology theories
  13. Thom classes

Key Result

Proposition 2.3

MR4480537 Let $f:S \to T$ be a finite étale morphism and $X$ be an $S$-scheme. Then we have

Theorems & Definitions (20)

  • Example 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3: MR4949893
  • proof
  • Remark 4.1
  • ...and 10 more