The special fiber of the motivic deformation of the stable homotopy category is algebraic

Zhouli Xu (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.; and Department of Mathematics, University of California San Diego, La Jolla, Calif., U.S.A.)

Abstract

For each prime $p$, we define a $t$‑structure on the category $\,\widehat{\!S^{0,0}}/\tau\text{-}\mathbf{Mod}_{\mathrm{harm}}^b$ of harmonic $\mathbb{C}$-motivic left-module spectra over $\,\widehat{\!S^{0,0}}/\tau$, whose MGL-homology has bounded Chow–Novikov degree, such that its heart is equivalent to the abelian category of $p$‑completed $\mathrm{BP}_*\mathrm{BP}$-comodules that are concentrated in even degrees. We prove that $\,\widehat{\!S^{0,0}}/\tau\text{-} \mathbf{Mod}_{\mathrm{harm}}^b$ is equivalent to $\mathcal{D}^b({\mathrm{BP}_*\mathrm{BP}\text{-}\mathbf{Comod}}^{\mathrm{ev}})$ as stable $\infty$-categories equipped with $t$‑structures.

As an application, for each prime $p$, we prove that the motivic Adams spectral sequence for $\,\widehat{\!S^{0,0}}/\tau$, which converges to the motivic homotopy groups of $\,\widehat{\!S^{0,0}}/\tau$, is isomorphic to the algebraic Novikov spectral sequence, which converges to the classical Adams–Novikov $E_2$-page for the sphere spectrum $\,\widehat{\!S^0}$. This isomorphism of spectral sequences allows Isaksen and the second and third authors to compute the stable homotopy groups of spheres at least to the $90$-stem, with ongoing computations into even higher dimensions.

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Received 25 October 2018

Received revised 29 January 2020

Accepted 13 May 2020

Published 2 July 2021