The special fiber of the motivic deformation of the stable homotopy category is algebraic
The special fiber of the motivic deformation of the stable homotopy category is algebraic
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稳定同伦范畴的动机变形的特殊纤维是代数的
DOI：
10.4310/acta.2021.v226.n2.a2
复制
发表时间：
2018-09
期刊：
影响因子：
3.7
通讯作者：
Zhouli Xu
中科院分区：
文献类型：
--
作者：
Bogdan Gheorghe;Guozhen Wang;Zhouli Xu
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.
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.
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DOI：
10.1090/conm/729/14699
发表时间：
2019
期刊：
Homotopy Theory: Tools and
Applications
影响因子：
--
作者：
Zhouli Xu
通讯作者：
Zhouli Xu
DOI：
10.1016/j.aim.2017.01.023
发表时间：
2016-03
期刊：
arXiv: Algebraic Topology
影响因子：
--
作者：
Irakli Patchkoria
通讯作者：
Irakli Patchkoria
影响因子：
1.7
作者：
Daniel Christensen
通讯作者：
Daniel Christensen
影响因子：
--
作者：
R. Bruner
通讯作者：
R. Bruner
DOI：
10.1017/cbo9781139107075.010
发表时间：
2008-07
期刊：
arXiv: Algebraic Topology
影响因子：
--
作者：
S. Schwede
通讯作者：
S. Schwede