Hovey-Strickland finite-generation question: are the homotopy groups of the $K(n)$-local sphere finitely generated over $\mathbb Z_p$?

Theorem 1.1: for every prime $p$, height $n\ge1$ and integer $t$, $\pi_t L_{K(n)}S_p^\wedge$ is a finitely generated $\mathbb Z_p$-module; equivalently $\pi_t L_{K(n)}F$ is finitely generated for every finite $p$-local spectrum $F$. Combined with the rational computation of Barthel-Schlank-Stapleton-Weinstein this fixes the free ranks. The key step is finiteness of continuous cohomology of the stabilizer with mod $p$ Lubin-Tate coefficients. It gives no bound on the number of generators or the torsion as $p$, $n$, $t$ vary, computes no torsion, and does not extend to all invertible $K(n)$-local spectra (known to fail at height two).

Theorem 1.1: for every prime $p$, height $n\ge1$ and integer $t$, $\pi_t L_{K(n)}S_p^\wedge$ is a finitely generated $\mathbb Z_p$-module; equivalently $\pi_t L_{K(n)}F$ is finitely generated for every finite $p$-local spectrum $F$. Combined with the rational computation of Barthel-Schlank-Stapleton-Weinstein this fixes the free ranks. The key step is finiteness of continuous cohomology of the stabilizer with mod $p$ Lubin-Tate coefficients. It gives no bound on the number of generators or the torsion as $p$, $n$, $t$ vary, computes no torsion, and does not extend to all invertible $K(n)$-local spectra (known to fail at height two).