# 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