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

@article{Gheorghe2021TheSF, 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={Acta Mathematica}, year={2021} }

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…

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## References

SHOWING 1-10 OF 84 REFERENCES

### The Motivic Cofiber of $\tau$

- Mathematics
- 2017

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 \colon…

### Some extensions in the Adams spectral sequence and the 51–stem

- MathematicsAlgebraic & Geometric Topology
- 2018

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 $\mathbb{Z}/8\oplus\mathbb{Z}/8\oplus\mathbb{Z}/2$.…

### K-theoretic obstructions to bounded t-structures

- MathematicsInventiones mathematicae
- 2019

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 main…

### The triviality of the 61-stem in the stable homotopy groups of spheres

- Mathematics
- 2016

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,…

### S-modules in the category of schemes

- Mathematics
- 2003

Introduction Preliminaries Coordinate-free spectra Coordinatized prespectra Comparison with coordinatized spectra The stable simplicial model structure The $\mathbb{A}^1$-local model structure…

### The strong Kervaire invariant problem in dimension 62

- Mathematics
- 2014

Using a Toda bracket computation $\langle \theta_4, 2, \sigma^2\rangle$ due to Daniel C. Isaksen [11], we investigate the $45$-stem more thoroughly. We prove that $\theta_4^2=0$ using a $4$-fold Toda…

### 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 a…

### Homotopy theory of comodules over a Hopf algebroid

- Mathematics
- 2003

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 for…

### Synthetic spectra and the cellular motivic category

- Mathematics
- 2018

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 SynE…

### A UNIVERSALITY THEOREM FOR VOEVODSKY'S ALGEBRAIC COBORDISM SPECTRUM

- Mathematics
- 2007

An algebraic version of a theorem of Quillen is proved. More precisely, for a regular Noetherian scheme S of finite Krull dimension, we consider the motivic stable homotopy category SH(S) of P 1…