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

We define the Chow $t$-structure on $\infty$-category of motivic spectra $\mathcal{SH}(k)$ over an arbitrary base field $k$. identify heart this $\mathcal{SH}(k)^{c\heartsuit}$ when exponential characteristic $k$ is inverted. Restricting to cellular subcategory, we $\mathcal{SH}(k)^{\mathrm{cell},c\heartsuit}$ as category even graded $\mathrm{MU}_{2*}\mathrm{MU}$-comodules. Furthermore, show that modules truncated sphere spectrum $\mathbbm{1}_{c=0}$ algebraic. Our results generalize ones in Gheorghe--Wang--Xu three aspects: integral results; all fields other than just $\mathbb{C}$ entire $\mathcal{SH}(k)$, rather a subcategory containing only certain objects. also discuss strategy for computing stable homotopy groups ($p$-completed) spheres using Postnikov--Whitehead tower associated and Adams spectral sequences