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

TL;DR

This paper introduces the Chow t-structure on the motivic spectra category over any base field, identifies its heart in various cases, and discusses methods for computing motivic stable homotopy groups.

Contribution

It generalizes previous results to integral coefficients, all base fields, and the entire motivic spectra category, providing new tools for motivic homotopy theory.

Findings

Identified the heart of the Chow t-structure over arbitrary fields.

Connected the Chow heart to even graded MU_{2*}MU-comodules.

Proposed a strategy for computing motivic stable homotopy groups.

Abstract

We define the Chow $t$-structure on the $\infty$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k)^{c\heartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $SH(k)^{cell, c\heartsuit}$ as the category of even graded $MU_{2*}MU$-comodules. Furthermore, we show that the $\infty$-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 $\infty$-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 $k$.