Galois reconstruction of Artin-Tate ℝ-motivic spectra

2020/10/20 by Robert Burklund, Jeremy Hahn, Burklund, Robert +3 · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.AG #math.AT #math.KT

paper · pdf · doi:10.48550/arxiv.2010.10325

openalex publication_date 2020/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We explain how to reconstruct the category of Artin-Tate ℝ-motivic spectra as a deformation of the purely topological C₂-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of C₂-equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of τ philosophy that has revolutionized classical stable homotopy theory. A key observation is that the Artin-Tate subcategory of ℝ-motivic spectra is easier to understand than the previously studied cellular subcategory. In particular, the Artin-Tate category contains a variant of the τ map, which is a feature conspicuously absent from the cellular category.

Citations

• On the nonexistence of elements of Kervaire invariant one
• Galois equivariance and stable motivic homotopy theory
• Fixed Point Free Conjugations on Complex Manifolds
• The six operations in equivariant motivic homotopy theory
• Synthetic spectra and the cellular motivic category
• On the boundaries of highly connected, almost closed manifolds
• Scheiderer motives and equivariant higher topos theory
• η-periodic motivic stable homotopy theory over fields
• The spectrum of derived Mackey functors
• The special fiber of the motivic deformation of the stable homotopy category is algebraic

Cited by

• Moduli stack of oriented formal groups and cellular motivic spectra over \mathbf C
• Filtrations in ℂ-motivic stable homotopy theory

Related

• Moduli stack of oriented formal groups and cellular motivic spectra over \mathbf C
• The special fiber of the motivic deformation of the stable homotopy category is algebraic
• Real isotropic motivic spectra
• Introduction to the Category of Derived Motivic Spectra
• Motivic stable stems and Galois approximations of cellular motivic categories