English

# Item

ITEM ACTIONSEXPORT

Released

Journal Article

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

##### MPS-Authors
/persons/resource/persons269323

Gheorghe,  Bogdan
Max Planck Institute for Mathematics, Max Planck Society;

##### Fulltext (restricted access)
There are currently no full texts shared for your IP range.
##### Fulltext (public)

1809.09290.pdf
(Preprint), 777KB

##### Supplementary Material (public)
There is no public supplementary material available
##### Citation

Gheorghe, B., Wang, G., & Xu, Z. (2021). The special fiber of the motivic deformation of the stable homotopy category is algebraic. Acta Mathematica, 226(2), 319-407. doi:10.4310/ACTA.2021.v226.n2.a2.

Cite as: https://hdl.handle.net/21.11116/0000-0009-EF12-6
##### 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 $BP_*BP$-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 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.