The Chow t-structure on the ∞-category of motivic spectra

2022/03/01 by Tom Bachmann, Hana Jia Kong, Guozhen Wang +1
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory

paper · doi:10.4007/annals.2022.195.2.5

Abstract

We define the Chow t-structure on the ∞-category of motivic spectra SH(k) over an arbitrary base field k. We identify the heart of this t-structure SH(k)^c\heartsuit when the exponential characteristic of k is inverted. Restricting to the cellular subcategory, we identify the Chow heart SH(k)^{cell,c\heartsuit} as the category of even graded MU_2*MU-comodules. Furthermore, we show that the ∞-category of modules over the Chow truncated sphere spectrum \mathbbm1_c=0 is algebraic. Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: to integral results; to all base fields other than just ℂ to the entire ∞-category of motivic spectra SH(k), rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field k using the Postnikov--Whitehead tower associated to the Chow t-structure and the motivic Adams spectral sequences over k.

Citations

• The special fiber of the motivic deformation of the stable homotopy category is algebraic

Related

• The Chow t-structure on the ∞-category of motivic spectra
• The special fiber of the motivic deformation of the stable homotopy category is algebraic
• Motivic stable stems and Galois approximations of cellular motivic categories
• Synthetic spectra and the cellular motivic category
• Galois reconstruction of Artin-Tate ℝ-motivic spectra