# On \'etale motivic spectra and Voevodsky's convergence conjecture

@inproceedings{Bachmann2020OnM, title={On \'etale motivic spectra and Voevodsky's convergence conjecture}, author={Tom Bachmann and Elden Elmanto and Paul arne Ostvaer}, year={2020} }

We prove a new convergence result for the slice spectral sequence, following work by Levine and Voevodsky. This verifies a derived variant of Voevodsky’s conjecture on convergence of the slice spectral sequence. This is, in turn, a necessary ingredient for our main theorem: a Thomason-style étale descent result for the Bott-inverted motivic sphere spectrum, which generalizes and extends previous étale descent results for special examples of motivic cohomology theories. Combined with first…

## References

SHOWING 1-10 OF 79 REFERENCES

Convergence of Voevodsky's Slice Tower

- Mathematics
- 2013

We consider Voevodsky's slice tower for a finite spectrum E in the motivic stable homotopy category over a perfect field k. In case k has finite cohomological dimension, we show that the slice tower…

Bloch-Kato Conjecture and Motivic Cohomology with Finite Coefficients

- Mathematics
- 2000

In this paper we show that the Beilinson-Lichtenbaum Conjecture which describes motivic cohomology of (smooth) varieties with finite coefficients is equivalent to the Bloch-Kato Conjecture, relating…

The homotopy groups of the η-periodic motivic
sphere spectrum

- Mathematics
- 2019

We compute the homotopy groups of the {\eta}-periodic motivic sphere spectrum over a finite-dimensional field k with characteristic not 2 and in which -1 a sum of four squares. We also study the…

Inverting the Motivic Bott Element

- Mathematics
- 2000

We prove a version for motivic cohomology of Thomason’s theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth…

TECHNIQUES OF LOCALIZATION IN THE THEORY OF ALGEBRAIC CYCLES

- Mathematics
- 1999

We extend the localization techniques of Bloch to simplicial spaces. As applications, we give an extension of Bloch’s localization theorem for the higher Chow groups to schemes of finite type over a…

$\eta$-periodic motivic stable homotopy theory over fields

- Mathematics
- 2020

Over any field of characteristic not 2, we establish a 2-term resolution of the $\eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is…

Perfection in motivic homotopy theory

- MathematicsProceedings of the London Mathematical Society
- 2019

We prove a topological invariance statement for the Morel–Voevodsky motivic homotopy category up to inverting exponential characteristics of residue fields. This implies in particular that SH1p of…

Hyperdescent and étale K-theory

- Mathematics
- 2019

We study the etale sheafification of algebraic K-theory, called etale K-theory. Our main results show that etale K-theory is very close to a noncommutative invariant called Selmer K-theory, which is…