On conjectures of Hovey–Strickland and Chai

Abstract

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>n2+n. Finally, we deduce consequences of our results for descent of Balmer spectra.