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

  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},
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.
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
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.
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
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
<p>We show that the stable homotopy groups of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="" alttext="upper C 2"> <mml:semantics> Expand
Nilpotence in normed MGL-modules
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
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
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


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
Some extensions in the Adams spectral sequence and the 51-stem
We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2-primary part of $\pi_{51}$ is $\mathbb{Z}/8\oplus\mathbb{Z}/8\oplus\mathbb{Z}/2$.Expand
K-theoretic obstructions to bounded t-structures
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
The triviality of the 61-stem in the stable homotopy groups of spheres
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
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
Homotopy theory of comodules over a Hopf algebroid
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
Synthetic spectra and the cellular motivic category
To any Adams-type homology theory we associate a notion of a synthetic spectrum, this is a spherical sheaf on the site of finite spectra with projective E-homology. We show that the ∞-category SynEExpand