Morava K-theory and Filtrations by Powers

TL;DR

This paper establishes the convergence of a Morava K-theory-based Adams spectral sequence, relates it to Lubin-Tate filtrations, and computes key limits, advancing understanding of chromatic homotopy theory.

Contribution

It introduces a new spectral sequence connecting K-theory homology to derived functors of completion, and relates it to Lubin-Tate filtrations using Miller squares.

Findings

Proves convergence of the Adams spectral sequence based on Morava K-theory.

Constructs a spectral sequence relating K-local sphere homology to derived functors of completion.

Computes the zeroth limit at all primes and heights.

Abstract

We prove the convergence of the Adams spectral sequence based on Morava K-theory and relate it to the filtration by powers of the maximal ideal in the Lubin-Tate ring through a Miller square. We use the filtration by powers to construct a spectral sequence relating the homology of the K-local sphere to derived functors of completion and express the latter as cohomology of the Morava stabilizer group. As an application, we compute the zeroth limit at all primes and heights.