On the motivic spectral sequence

TL;DR

This paper demonstrates the equivalence between the Grayson tower and the Voevodsky slice tower for motivic K-theory, resolving Suslin's problem by linking two key spectral sequences in algebraic geometry.

Contribution

It establishes an isomorphism between the Grayson and Voevodsky spectral sequences, extending the Grayson tower to bispectra and confirming their equivalence.

Findings

The Grayson tower for K-theory is isomorphic to the slice tower of S^1-spectra.

The Grayson motivic spectral sequence matches the Voevodsky motivic spectral sequence.

The results affirm Suslin's problem for these spectral sequences.

Abstract

It is shown that the Grayson tower for $K$-theory of smooth algebraic varieties is isomorphic to the slice tower of $S^1$-spectra. We also extend the Grayson tower to bispectra and show that the Grayson motivic spectral sequence is isomorphic to the motivic spectral sequence produced by the Voevodsky slice tower for the motivic $K$-theory spectrum $KGL$. This solves Suslin's problem for these two spectral sequences in the affirmative.