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

We define the Chow $t$-structure on the $\mathrm{\infty }$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k{)}^{c\mathrm{♡}}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $SH(k{)}^{cell,c\mathrm{♡}}$ as the category of even graded $M{U}_{2\ast }MU$-comodules. Furthermore, we show that the $\mathrm{\infty }$-category of modules over the Chow truncated sphere spectrum 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 $\mathrm{\infty }$-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 tower associated to the Chow $t$-structure and the motivic Adams spectral sequences over $k$.