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 MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to the abelian category of $p$-completed $B{P}_{\ast }BP$-comodules that are concentrated in even degrees. We prove that $\widehat{S^{0,0}}/\tau \text{-}{\mathbf{Mod}}_{harm}^b$ is equivalent to ${\mathcal{D}}^b({{BP}_{\ast }BP\text{-}\mathbf{Comod}}^{ev})$ 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 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.

arXiv commentAccepted version, 85 pages

References in corpus (7)

• Higher Topos Theory
• An exact sequence for Milnor's K-theory with applications to quadratic forms
• Algebraic cobordism
• From algebraic cobordism to motivic cohomology
• Algebraic versus topological triangulated categories
• On exact infinity-categories and the Theorem of the Heart
• Some extensions in the Adams spectral sequence and the 51-stem

Cited by in corpus (4)

• Stable homotopy groups of spheres
• Detecting exotic spheres in low dimensions using coker J
• An extension in the Adams spectral sequence in dimension 54
• Topological models for stable motivic invariants of regular number rings