Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra

Abstract

We explain how to reconstruct the category of Artin-Tate R-motivic spectra as a deformation of the purely topological C 2 -equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of C 2 -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. Contents 1. Introduction 1 2. Constructing the element ta 17 3. The ta-local category 19 4. Galois reconstruction 24 5. Modules over the cofiber of ta 30 6. The Chow t-structure and ν R 44 7. The a-local category 53 8. Completions 55 9. Odd primes 60 10. Examples and computations 61 Appendix A. Recollections on compact rigid generation 65 Appendix B. Recollections on filtered objects 68 Appendix C. A machine for deforming homotopy theories 70 References 75

Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf C$

“…This notion was introduced, and thoroughly studied in analogy with other homotopytheoretic incarnations of cellularity, in [DI05]. On the other hand, much of the motivic literature, including [BHS20b], prefers the more traditional algebro-geometric term Tate motivic spectra. This has to do with the fact that the spheres S 0,n for n ∈ Z, which generate Sp cell C under limits, colimits and extensions, encode in motivic cohomology analogous structure to Tate twists in étale cohomology.…”

“…τ = 0, is the ∞-category of ind-coherent sheaves IndCoh(M ♡ FG ). Hence the name τ -deformation picture; see also [BHS20b,Subsection 1.3,p. 10] for further discussion on this perspective.…”

“…Remark 2. With the exception of [BHS20b], most of the literature on this subject does not make explicit mention of ind-coherent sheaves, instead preferring to view the special fiber at τ = 0 as Hovey's stable category of MU * MU-comodules. Since the two ∞-categories are equivalent by [BHV18,Proposition 5.40], this is merely an aesthetic difference.…”

“…The τ -deformation picture has been extended in many directions: in [BHS20b] to work over the base R and relating to C 2 -equivariant spectra, in [Pst18] to the integral (i.e. non-p-local) setting via synthetic spectra, and in [BKWX21] to an integral setting over any field k via the Chow t-structure.…”

“…non-p-local) setting via synthetic spectra, and in [BKWX21] to an integral setting over any field k via the Chow t-structure. In [GIKR18], an alternative approach to proving Theorem 1 is given (also followed by [BHS20b]), which relies on filtered spectra.…”