On conjectures of Hovey--Strickland and Chai

We prove the height two case of a conjecture of Hovey and Strickland that provides a $K(n)$-local analogue of the Hopkins--Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by Chai. We then use the Gross--Hopkins period map to verify Chai's Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava $E$-theory is coherent, and that every finitely generated Morava module can be realized by a $K(n)$-local spectrum as long as $2p-2>n^2+n$. Finally, we deduce consequences of our results for descent of Balmer spectra.