Chromatic homotopy is algebraic when $p > n^{2}+n+1$

Abstract

We show that if E is a p-local Landweber exact homology theory of height n and p > n 2 + n + 1, then there exists an equivalence hSp E ≃ hD(E * E) between homotopy categories of E-local spectra and differential E * E-comodules, generalizing Bousfield's and Franke's results to heights n > 1. 1.4. Acknowledgements 5 2. Realization of comodules 5 3. Derived ∞-category of differential comodules 9 4. Goerss-Hopkins theory 10 5. The Bousfield adjunction 14 6. An algebraic model for the E-local homotopy category 16 Appendix A. Connectivity results for undercategories 20 Appendix B. The homotopy category of a stable ∞-category 21 References 23

Adams spectral sequences and Franke's algebraicity conjecture

“…where k " 2p ´2 ´n2 ´n. This is an extension of a result from second author's thesis, which proved an equivalence of homotopy categories under the weaker bound 2p ´2 ą 2pn 2 `nq, as well as the classical work of Bousfield at n " 1 [21], [77]. The current work improves the range in which equivalences exist to the one predicted by Franke.…”

“…Our general approach to Franke's conjecture follows the strategy from the second's author thesis [77]: one wants to embed C into a suitable "derived 8-category" and apply Goerss-Hopkins theory [43], [79]. In the case of Adams-type homology theories on spectra, a suitable derived 8-category is given by synthethic spectra [78].…”

Morava K-theory and Filtrations by Powers

“…Proof. This is proven under the assumption 2p − 2 > 2(n 2 + n) in the second author's thesis [Pst18a]. The bound was improved to the one given above in joint work with Patchkoria [PP21].…”

“…To provide some context, recall that it is known that at large primes there exists an equivalence hSp E ≃ hD per (E * E) between the homotopy categories of E-local spectra and differential comodules. Under this equivalence, the limit lim ← − E * /m k in the derived ∞-category corresponds to the K-local sphere S 0 K [Pst18a]; thus, the derived completion of E * can be thought of as the algebraic analogue of the K-local sphere.…”

Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf C$

“…On the other hand, [Pst18] has proposed another explanation of the τ -deformation picture, by introducing the algebraic notion of synthetic spectra. These have since found other applications, for instance to asymptotic chromatic algebraicity [Pst21], Goerss-Hopkins obstruction theory [PV21], and questions of manifold geometry in [BHS19], [BHS20a], and have been expanded in scope in [PP21]. Using the same filtered module spectra technique, which we use in the p-complete setting to prove Theorem 3, we obtain in Subsection 1.7 an integral comparison with synthetic spectra.…”