On conjectures of Hovey--Strickland and Chai

TL;DR

This paper proves a key case of a conjecture relating to $K(n)$-local spectra, using arithmetic geometry and the Gross--Hopkins map, with implications for the structure of Morava $E$-theory and Balmer spectra.

Contribution

It establishes the height two case of Hovey and Strickland's conjecture, linking it to arithmetic geometry and verifying Chai's Hope at height two for all primes.

Findings

Proved the height two case of the Hovey--Strickland conjecture.

Showed the coherence of the completed cooperations ring for Morava $E$-theory.

Established that finitely generated Morava modules can be realized by $K(n)$-local spectra under certain conditions.

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 $2p-2>n^2+n$. Finally, we deduce consequences of our results for descent of Balmer spectra.