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Motivic and Étale Spanier-Whitehead duality and the Becker-Gottlieb transfer

Gunnar Carlsson, Roy Joshua

Abstract

In this paper, we develop a theory of Becker-Gottlieb transfer based on Spanier-Whitehead duality that holds in both the motivic and étale settings for smooth quasi-projective varieties in as broad a context as possible: for example, for varieties over non-separably closed fields in all characteristics, and also for both the étale and motivic settings. In view of the fact that the most promising applications of the traditional Becker-Gottlieb transfer has been to torsors and Borel-style equivariant cohomology theories, we focus our applications to motivic cohomology theories for torsors as well as Borel-style equivariant motivic cohomology theories, both defined with respect to motivic spectra. We obtain several results in this direction, including a stable splitting in generalized motivic cohomology theories. Various further applications will be discussed in forthcoming papers.

Motivic and Étale Spanier-Whitehead duality and the Becker-Gottlieb transfer

Abstract

In this paper, we develop a theory of Becker-Gottlieb transfer based on Spanier-Whitehead duality that holds in both the motivic and étale settings for smooth quasi-projective varieties in as broad a context as possible: for example, for varieties over non-separably closed fields in all characteristics, and also for both the étale and motivic settings. In view of the fact that the most promising applications of the traditional Becker-Gottlieb transfer has been to torsors and Borel-style equivariant cohomology theories, we focus our applications to motivic cohomology theories for torsors as well as Borel-style equivariant motivic cohomology theories, both defined with respect to motivic spectra. We obtain several results in this direction, including a stable splitting in generalized motivic cohomology theories. Various further applications will be discussed in forthcoming papers.

Paper Structure

This paper contains 34 sections, 25 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Basic Model category framework for simplicial presheaves
  3. Simplicial presheaves on ${\rm Sm}_{\it k}$
  4. Pointed simplicial presheaves
  5. The motivic model structure on Nisnevich presheaves
  6. The injective model structures
  7. The motivic model structure on étale simplicial presheaves: $\rm {\bf Spc}_*(\rm S_{et})$
  8. Pointed equivariant simplicial presheaves
  9. Model structures on equivariant simplicial presheaves
  10. The $\rm G$-equivariant sheaves $\rm G/H_+$, $\rm H \, {\boldsymbol\varepsilon} \, \mathbf W$
  11. Finitely presented objects
  12. A Key observation
  13. Categories of spectra
  14. Equivariant spectra
  15. Derived functors of $\wedge$, the internal ${\mathcal{H}}om$ and the dual ${\rm D}$ for equivariant spectra
  16. ...and 19 more sections

Key Result

Theorem \oldthetheorem

(Dualizability)

Theorems & Definitions (73)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 63 more