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 $\mathbb{1}_{c=0}$ is algebraic.
Our results generalize the ones in Gheorghe–Wang–Xu in three aspects: to integral results; to all base fields other than just $\Bbb{C}$; 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 $k$.
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issn = {0003-486X},
mrclass = {14F42 (19D45)},
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@MISC{wendt2010more,
author = {Wendt, Matthias},
title = {More examples of motivic cell structures},
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zblnumber = {},
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@ARTICLE{WilsonOst,
author = {Wilson, Glen Matthew and {\O}stv{æ}r, Paul Arne},
title = {Two-complete stable motivic stems over finite fields},
journal = {Algebr. Geom. Topol.},
fjournal = {Algebraic \& Geometric Topology},
volume = {17},
year = {2017},
number = {2},
pages = {1059--1104},
issn = {1472-2747},
mrclass = {14F42 (18G15 55T15)},
mrnumber = {3623682},
mrreviewer = {Daniel C. Isaksen},
doi = {10.2140/agt.2017.17.1059},
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author = {Xu, Zhouli},
title = {Conference talk ``{C}omputing stable homotopy groups of spheres" at {H}omotopy {T}heory: {T}ools and {A}pplications},
note = {University of Illinois at Urbana-Champaign},
year = {2017},
zblnumber = {},
}