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arXiv · 2012.02687

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

Tom Bachmann·Hana Jia Kong·Guozhen Wang·Zhouli Xu
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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 $k$.

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Tom Bachmann, Hana Jia Kong, Guozhen Wang, Zhouli Xu. 2020-12-04. The Chow $t$-structure on the $\infty$-category of motivic spectra. https://arxiv.org/abs/2012.02687

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