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

The Spk,n–local stable homotopy category

“…k;n ; this is a consequence of the characterization of compact objects given in Theorem 3.8, and that D 0 D Sp dual k;n .Hovey and Strickland conjecture that in the case k D n these exhaust the thick-tensor ideals of Sp dual K.n/ . This has been investigated in detail in[10]. We conjecture this holds more generally in Sp k;n .…”

“…For this, we recall that Hovey and Strickland have conjectured a description of the Balmer spectrum Spc.Sp dual K.n/ / of dualizable objects in K.n/-local spectra [32, page 61]. This was investigated by the author, along with Barthel and Naumann, in [10]. This admits a natural generalization to Sp dual k;n .…”

“…One could also ask for a classification of the thick tensorideals of dualizable objects in Sp k;n , or equivalently a computation of the Balmer spectrum Spc.Sp dual k;n / (which is well defined by Lemma 5.12). Based on a conjecture of Hovey and Strickland, the author, along with Barthel and Naumann, investigated Spc.Sp dual K.n/ / in detail in[10], showing that the Hovey-Strickland conjecture holds when n D 1 and 2, and that in general it is implied by a hope of Chai in arithmetic geometry. In this section, we make some general comments regarding Spc.Sp dual k;n /.Remark 5.13 The following full subcategories were considered in the case k D n by Hovey and Strickland [32, Definition 12.14].…”

“…n , then C D C j for some j such that 0 Ä j Ä n C 1.Remark 3.14 This result can be restated in terms of the Balmer spectrum of Sp ! with topology determined by the closure operator fC j g D fC i j i j g. This is in fact equivalent to Theorem 3.13, essentially by the same argument as in[10, Proposition 3.5].For a thick subcategory J , we define supp.J / D S X 2J supp.X /. Then, Balmer's classification result[3, Theorem 4.10] shows that there is a bijection fthick subcategories of Sp !…”

Cosupport in tensor triangular geometry

“…We will not pursue this class of examples in this work, but it would be an interesting support theory to study, e.g., for the K(n)-local category. Some work in this direction has been carried out in [BHN22].…”