Algebraic slice spectral sequences

TL;DR

This paper constructs a framework linking different motivic spectral sequences, providing new computational tools for motivic spectra like algebraic K-theory and Brown-Peterson spectra, and relates them to known spectral sequences.

Contribution

It introduces a square of spectral sequences connecting the effective slice and motivic Adams spectral sequences for key motivic spectra, enhancing computational understanding.

Findings

The square of spectral sequences is constructed for several motivic spectra.

The $ ho$-Bockstein spectral sequence determines the $ ext{R}$-motivic effective slice spectral sequence.

Applications to Hill-Hopkins-Ravenel slice spectral sequences are obtained.

Abstract

For certain motivic spectra, we construct a square of spectral sequences relating the effective slice spectral sequence and the motivic Adams spectral sequence. We show the square can be constructed for connective algebraic K-theory, motivic Morava K-theory, and truncated motivic Brown-Peterson spectra. In these cases, we show that the $\mathbb{R}$-motivic effective slice spectral sequence is completely determined by the $\rho$-Bockstein spectral sequence. Using results of Heard, we also obtain applications to the Hill-Hopkins-Ravenel slice spectral sequences for connective Real K-theory, Real Morava K-theory, and truncated Real Brown-Peterson spectra.