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

@article{Gheorghe2018TheSF,
title={The special fiber of the motivic deformation of the stable homotopy category is algebraic},
author={Bogdan Gheorghe and Guozhen Wang and Zhouli Xu},
journal={arXiv: Algebraic Topology},
year={2018}
}
• Published 2018
• Mathematics
• arXiv: Algebraic Topology
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 $\mathcal{D}^b({{BP}_*{BP… Expand The Chow$t$-structure on motivic spectra. • Mathematics • 2020 We define the Chow$t$-structure on the$\infty$-category of motivic spectra$SH(k)$over an arbitrary base field$k$. We identify the heart of this$t$-structure$SH(k)^{c\heartsuit}$when theExpand Galois reconstruction of Artin-Tate$\mathbb{R}$-motivic spectra • Mathematics • 2020 We explain how to reconstruct the category of Artin-Tate$\mathbb{R}$-motivic spectra as a deformation of the purely topological$C_2$-equivariant stable category. The special fiber of thisExpand Complex motivic$kq$-resolutions. • Mathematics • 2019 We analyze the$kq$-based motivic Adams spectral sequence over the complex numbers, where$kq$is the very effective cover of Hermitian K-theory defined over$\mathbb{C}$by Isaksen-Shkembi and overExpand Real motivic and C2‐equivariant Mahowald invariants The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. In \cite{Qui17}, we studied an analog of thisExpand C_2-equivariant and R-motivic stable stems, II • Mathematics • 2020 We show that the$C_2$-equivariant and$\mathbb{R}$-motivic stable homotopy groups are isomorphic in a range. This result supersedes previous work of Dugger and the third author. 𝐶₂-equivariant and ℝ-motivic stable stems II • Mathematics • 2020 <p>We show that the stable homotopy groups of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 2"> <mml:semantics> Expand Nilpotence in normed MGL-modules • Mathematics • 2019 We establish a motivic version of the May Nilpotence Conjecture: if E is a normed motivic spectrum that satisfies$E \wedge HZ \simeq 0$, then also$E \wedge MGL \simeq 0$. In words, motivic homologyExpand Stable homotopy groups of spheres • Medicine, Mathematics • Proceedings of the National Academy of Sciences • 2020 The groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4 are determined, which are more heavily on machine computations than previous methods and is therefore less prone to error. Expand The homotopy of C-motivic modular forms A C-motivic modular forms spectrum mmf has recently been constructed. This article presents detailed computational information on the Adams spectral sequence for mmf. This information is essentialExpand C-motivic modular forms • Mathematics • 2018 We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations. We compute the Steenrod algebra in this context, and weExpand #### References SHOWING 1-10 OF 95 REFERENCES The Motivic Cofiber of$\tau$Consider the Tate twist$\tau \in H^{0,1}(S^{0,0})$in the mod 2 cohomology of the motivic sphere. After 2-completion, the motivic Adams spectral sequence realizes this element as a map$\tau \colonExpand
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Schlichting conjectured that the negative K-groups of small abelian categories vanish and proved this for noetherian abelian categories and for all abelian categories in degree $$-1$$-1. The mainExpand
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We prove that the 2-primary $\pi_{61}$ is zero. As a consequence, the Kervaire invariant element $\theta_5$ is contained in the strictly defined 4-fold Toda bracket $\langle 2, \theta_4, \theta_4,Expand The derived category of complex periodic K-theory localized at an odd prime We prove that for an odd prime$p$, the derived category$\mathcal{D}(KU_{(p)})$of the$p$-local complex periodic$K$-theory spectrum$KU_{(p)}$is triangulated equivalent to the derived category ofExpand S-modules in the category of schemes Introduction Preliminaries Coordinate-free spectra Coordinatized prespectra Comparison with coordinatized spectra The stable simplicial model structure The$\mathbb{A}^1$-local model structureExpand On exact$\infty$-categories and the Theorem of the Heart The new homotopy theory of exact$\infty$-categories is introduced and employed to prove a Theorem of the Heart for algebraic$K$-theory (in the sense of Waldhausen). This implies a new compatibilityExpand On localization sequences in the algebraic K-theory of ring spectra • Mathematics • 2014 We identify the$K$-theoretic fiber of a localization of ring spectra in terms of the$K\$-theory of the endomorphism algebra spectrum of a Koszul-type complex. Using this identification, we provide aExpand
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Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework forExpand
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