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Motivic stable stems and Galois approximations of cellular motivic categories

Tom Bachmann, Robert Burklund, Zhouli Xu

Abstract

We reconstruct (appropriately completed) categories of cellular motivic spectra over fields of small cohomological dimension in terms of only their absolute Galois groups. As our main application, we determine the motivic stable stems (away from the characteristic) of almost all fields.

Motivic stable stems and Galois approximations of cellular motivic categories

Abstract

We reconstruct (appropriately completed) categories of cellular motivic spectra over fields of small cohomological dimension in terms of only their absolute Galois groups. As our main application, we determine the motivic stable stems (away from the characteristic) of almost all fields.

Paper Structure

This paper contains 63 sections, 118 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Invitation: the motivic stable stems
  3. Sketch proof of a special case
  4. Galois approximation
  5. Quillen--Lichtenbaum properties and étale algebraic cobordism
  6. Reverse-linear overview
  7. Notation and conventions
  8. Acknowledgements
  9. Preliminaries
  10. Graded objects
  11. Pro-objects
  12. Unstable $\ell$-completion
  13. Spaces
  14. General case
  15. $\Sigma$-presheaves
  16. ...and 48 more sections

Key Result

Theorem 1.1

Let $k$ be a field of characteristic $\ne \ell$ containing all $\ell^n$-th roots of unity for all $n$. There is an isomorphism

Theorems & Definitions (301)

  • Theorem 1.1: see Theorem \ref{['thm:tensor-product-formula-easy']}
  • Remark 1.2
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 291 more