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

##### External Resource

https://dx.doi.org/10.4310/ACTA.2021.v226.n2.a2

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##### Fulltext (public)

1809.09290.pdf

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##### 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.

$\widehat{S^{0,0}}/\tau\text{-}\mathbf{Mod}_{harm}^b$ 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.