A descent spectral sequence for arbitrary K(n)-local spectra with explicit $E_2$-term

TL;DR

This paper constructs a new descent spectral sequence for arbitrary K(n)-local spectra, providing an explicit E_2-term as continuous cohomology of the Morava stabilizer group, enhancing understanding of K(n)-local homotopy groups.

Contribution

It introduces a descent spectral sequence with an explicit E_2-term for K(n)-local spectra, differing from the traditional E_n-Adams spectral sequence.

Findings

Spectral sequence converges to pi_*(L_{K(n)}(X))

E_2-term is explicitly given by continuous cohomology of G_n

Provides a more explicit computational tool for K(n)-local spectra

Abstract

Let n be any positive integer and p any prime. Also, let X be any spectrum and let K(n) denote the nth Morava K-theory spectrum. Then we construct a descent spectral sequence with abutment pi_*(L_{K(n)}(X)) and E_2-term equal to the continuous cohomology of G_n, the extended Morava stabilizer group, with coefficients in a certain discrete G_n-module that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)-local E_n-Adams spectral sequence for pi_*(L_{K(n)}(X)), whose E_2-term is not known to always be equal to a continuous cohomology group.