Algebraic chromatic homotopy theory for BP∗BP-comodules

Abstract

In this paper, we study the global structure of an algebraic avatar of the derived category of ind-coherent sheaves on the moduli stack of formal groups. In analogy with the stable homotopy category, we prove a version of the nilpotence theorem as well as the chromatic convergence theorem, and construct a generalized chromatic spectral sequence. Furthermore, we discuss analogs of the telescope conjecture and chromatic splitting conjecture in this setting, using the local duality techniques established earlier in joint work with Valenzuela.Contents Recall that, working in the p-complete setting, the motivic cohomology of a point over Spec(C) is isomorphic to Fp[τ ], where τ has bidegree (0,1), and that this gives rise to an essential map τ : S 0,−1 → S 0,0 .Theorem (The chromatic spectral sequence). For any spectra X, Y , there is a natural convergent spectral sequence E n,s,t 1 = Ext s,t BP Furthermore, if BP * Y satisfies the conditions of Section 1, then the spectral sequence converges to Ext BP * BP (BP * X, BP * Y ).By truncating the chromatic tower, we can also build a height n analog of the chromatic spectral sequence which, as a special case, recovers the truncated chromatic spectral sequence constructed by Hovey and Sadofsky [29, Theorem 5.1].As a concrete application of our results, we obtain the following transchromatic comparison between the E 2 -terms of the BP -Adams spectral sequence and the E-Adams spectral sequence at height n, see Corollary 7.19.Corollary. If X is a p-local bounded below spectrum such that BP * X has projective BP * -dimension pdim(BP * X) r, then the natural map Ext s BPis an isomorphism if s < n − r − 1 and injective for s = n − r − 1.A related result can be found in work of Goerss [20, Theorem 8.24], however the authors are unaware of a result of this generality in the literature.