The special fiber of the motivic deformation of the stable homotopy category is algebraic
Abstract
For each prime $p$, we define a $t$structure on the category $\widehat{S^{0,0}}/\tau\text{}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$motivic left module spectra over $\widehat{S^{0,0}}/\tau$, whose MGLhomology has bounded ChowNovikov degree, such that its heart is equivalent to the abelian category of $p$completed $BP_*BP$comodules that are concentrated in even degrees. We prove that $\widehat{S^{0,0}}/\tau\text{}\mathbf{Mod}_{harm}^b$ is equivalent to $\mathcal{D}^b({{BP}_*{BP}\text{}\mathbf{Comod}}^{ev})$ 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 AdamsNovikov $E_2$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 90stem, with ongoing computations into even higher dimensions.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1809.09290
 Bibcode:
 2018arXiv180909290G
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Algebraic Geometry;
 Mathematics  Category Theory;
 Mathematics  KTheory and Homology
 EPrint:
 Accepted version, 85 pages