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 $\mathcal{SH}(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $\mathcal{SH}(k)^{c\heartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $\mathcal{SH}(k)^{\mathrm{cell},c\heartsuit}$ as the category of even graded $\mathrm{MU}_{2*}\mathrm{MU}$-comodules. Furthermore, we show that the $\infty$-category of modules over the Chow truncated sphere spectrum $\mathbbm{1}_{c=0}$ 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 $\mathcal{SH}(k)$, 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 $k$.