The Hovey-Strickland conjecture: the thick tensor ideals of dualizable K(n)-local spectra

Theorem 1.2: for every prime $p$ and $n\ge1$ the thick tensor ideals of the dualizable $K(n)$-local category are exactly $\mathcal D_0\supsetneq\cdots\supsetneq\mathcal D_n\supsetneq\mathcal D_{n+1}=0$, so there are $n+2$ of them and each $\mathcal D_k$ does not depend on the chosen type-$k$ spectrum; a corollary identifies the Balmer spectrum with the $n+1$ proper members of the chain, with its specialization order. It is deduced from Theorem 1.1 (Chai's conjecture, a separate entry) by the known Barthel-Heard-Naumann implication. It concerns dualizable objects only and does not classify localizing or other subcategories of the whole $K(n)$-local category. Height two was already known.

Theorem 1.2: for every prime $p$ and $n\ge1$ the thick tensor ideals of the dualizable $K(n)$-local category are exactly $\mathcal D_0\supsetneq\cdots\supsetneq\mathcal D_n\supsetneq\mathcal D_{n+1}=0$, so there are $n+2$ of them and each $\mathcal D_k$ does not depend on the chosen type-$k$ spectrum; a corollary identifies the Balmer spectrum with the $n+1$ proper members of the chain, with its specialization order. It is deduced from Theorem 1.1 (Chai's conjecture, a separate entry) by the known Barthel-Heard-Naumann implication. It concerns dualizable objects only and does not classify localizing or other subcategories of the whole $K(n)$-local category. Height two was already known.