  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.
