Morava K-theory and Filtrations by Powers

Abstract

We prove the convergence of the Adams spectral sequence based on Morava Ktheory 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. Contents 1. Introduction Part 1. Morava K-theories and Milnor modules 2. Morava K-theories and their Hopf algebroids 3. Digression: Minimal Morava K-theory 4. Milnor modules 5. Cohomology 6. Milnor filtration Part 2. The K-based Adams spectral sequence 7. Convergence of the K-based Adams spectral sequence 8. The finite height Miller square 9. The K-based Adams spectral sequence at large primes 10. Height one K-based Adams spectral sequence at an odd prime Part 3. Homology of inverse limits and the algebraic chromatic splitting conjecture 11. The homology of inverse limits 12. The Morava E-homology of inverse limits 13. Digression: Cohomology of G n as Ext in comodules 14. Algebraic chromatic splitting conjecture 15. Cooperations at height one References