The Chow $t$-structure on the $\infty$-category of motivic spectra

Abstract

We define the Chow $t$-structure on the $\infty$-category of motivic spectra $\mathcal{S}\mathcal{H}(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $\mathcal{S}\mathcal{H}(k)^{c\heartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $\mathcal{S}\mathcal{H}(k)^{\mathrm{cell}, c\heartsuit}$ as the category of even graded $\mathrm{M}\mathrm{U}_{2*}\mathrm{M}\mathrm{U}$-comodules. Furthermore, we show that the $\infty$-category of modules over the Chow truncated sphere spectrum $\mathbb{l}_{c=0}$ is algebraic.

Our results generalize the ones in Gheorghe–Wang–Xu in three aspects: To integral results; To all base fields other than just $\mathbb{C}$; To the entire $\infty$-category of motivic spectra $\mathcal{S}\mathcal{H}(k)$, 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 $k$.

Authors

Tom Bachmann

Mathematisches Institut, LMU Munich, Munich, Germany

Hana Jia Kong

Department of Mathematics, The University of Chicago, Chicago, IL, USA

Current address:

School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA Guozhen Wang

Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China

Zhouli Xu

Department of Mathematics, University of California San Diego, La Jolla, CA, USA