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On conjectures of Hovey-Strickland and Chai. (English) Zbl 1515.14054

Summary: We prove the height two case of a conjecture of M. Hovey and N. P. Strickland [Morava \(K\)-theories and localisation. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0929.55010)] 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 C.-L. Chai’s Hope [Duke Math. J. 82, No. 3, 725–754 (1996; Zbl 0864.14028)] 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.

MSC:

14L05 Formal groups, \(p\)-divisible groups
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
11S31 Class field theory; \(p\)-adic formal groups
18G80 Derived categories, triangulated categories
55P42 Stable homotopy theory, spectra
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