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A topological model for cellular motivic spectra

Hadrian Heine

Abstract

For any motivic $\mathbb{E}_\infty$-ring spectrum $A$ we construct an equivalence $ρ$ between the $\infty$-category of cellular motivic $A$-module spectra and modules over an $\mathbb{E}_1$-algebra $Θ$ in $\mathbb{Z} $-graded spectra, under which the motivic grading corresponds to the $\mathbb{Z}$-grading. If the base is the complex numbers or if $A$ admits an $\mathbb{E}_\infty$-orientation, we refine the $\mathbb{E}_1$-algebra $Θ$ to an $\mathbb{E}_\infty$-algebra and $ρ$ to a symmetric monoidal equivalence. To capture the symmetric monoidal structure in the general situation, we lift $ρ$ to a symmetric monoidal equivalence to modules over an $\mathbb{E}_\infty$-algebra in $\mathcal{J} $-graded spectra that invert morphisms of $\mathcal{J}$, where $\mathcal{J}$ is the diagram category of Sagave-Schlichtkrull, a model for Quillen's localization of the groupoid of finite sets and bijections.

A topological model for cellular motivic spectra

Abstract

For any motivic -ring spectrum we construct an equivalence between the -category of cellular motivic -module spectra and modules over an -algebra in -graded spectra, under which the motivic grading corresponds to the -grading. If the base is the complex numbers or if admits an -orientation, we refine the -algebra to an -algebra and to a symmetric monoidal equivalence. To capture the symmetric monoidal structure in the general situation, we lift to a symmetric monoidal equivalence to modules over an -algebra in -graded spectra that invert morphisms of , where is the diagram category of Sagave-Schlichtkrull, a model for Quillen's localization of the groupoid of finite sets and bijections.

Paper Structure

This paper contains 10 sections, 30 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Notation and Terminology
  4. Acknowledgements
  5. Cellular objects
  6. Dualizable and tensor invertible objects
  7. Graded objects are dualizably generated
  8. A model for cellular objects
  9. A model for cellular motivic spectra
  10. Appendix

Key Result

Theorem 1.1

(Theorem mot) Let ${\mathrm S}$ be a base scheme, $n \in {\mathbb Z}$ and $A$ a motivic ${\mathbb E}_\infty$-ring spectrum over ${\mathrm S}$. There is a canonical equivalence

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.1
  • Example 2.3
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • ...and 58 more